Modeling and simulation of a continuous fluidized-bed dryer · 2017-12-15 ·...

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/modeling and simulation of a continuous fluldized-bed dryer/

by

YIMING CHEN

B.E. ChE. Zhejiang University, Hangzhou, China. 1982

A MASTER'S THESIS

submitted in partial fulfillment of the

requirements for the degree

MASTER OF SCIENCE

Department of Chemical Engineering

KANSAS STATE UNIVERSITY

Manhattan, Kansas

1986

Approved by:

2*X:Major Professor

Co-Major Professor

ate*•rf

TABLE OF CONTENTS

A112D2 1bS3DM Page

LISTS OF TABLES.

.

LISTS OF FIGURES.

ACKNOWLEDGEMENTS

.

CHAPTER I INTRODUCTION 1-1

CHAPTER II LITERATURE REVIEW

INTRODUCTION. . II-l

CLASSIFICATION OF FLUIDIZED-BED DRYERS II-l

Single-Stage Dryers II-2

Multiple-Stage Dryers II-2

Centrifugal Bed Dryers j 1 — 3

Vibrated Bed Dryers II—

1

PHENOMENA OF FLUIDIZATION n _ 4

Pressure Drop and Minimum Fluidization Ve 1 oc i t y . . . I I-

5

Columnar beds ..II-5

Tapered beds II-6

Centrifugal beds II-7

Vibrated beds ..II-9

Bed Expansion 11-10

Fluid Mechani st ical Strcture of the Bed 11-13

Formation of phases 11-13

Gas movement and gas mixing 11-15

Solids movement and solids mixing ..11-16

TRANSPORT PROCESSES ..11-19

Gas Interphase Exchange ..11-19

Gas-to-Partiole Heat Transfer

Mechanism

Correlations for overall coefficients

Bed-to-Surf ace Heat Transfer

Mechanism


Par t icl e-to-Gas Mass Transfer

Mechanism


DRYING CHARACTERISTICS OF SOLIDS

Constant Rate Drying Period

Falling Rate Drying Period

MODELING OF FLUIDIZED-BED DRYER

Batch Operation

Model proposed by Viswanathan and Rao

Model proposed by Hoebink

Continuous Operation

Model proposed by Palancz

NOTATION

LITERATURE CITED

APPENDIX A. DERIVATION OF EON. (87)

APPENDIX B. DERIVATION OF EON. (88)

APPENDIX C. DERIVATION OF EQN. (89)

APPENDIX D. DERIVATION OF EQN. (117)

APPENDIX E. DERIVATION OF EQN. (119)

APPENDIX F. DERIVATION OF EQN. (146)

APPENDIX G. DERIVATION OF EQN. (154)

11-21

11-21

11-21

11-23

11-23

11-24

11-30

11-30

11-31

11-36

11-36

11-37

11-43

11-43

11-44

11-55

11-69

11-69

11-73

11-78

11-85

11-86

11-88

11-90

11-91

11-93

11-95

APPENDIX H. DERIVATION OF EQN. (161)

APPENDIX H. DERIVATION OF EQN. (165).

APPENDIX J. DERIVATION OF EQN. (187),

CHAPTER III MODELING AND SIMULATION OF A CONTINOUS

FLUIDIZED-BED DRYER

MATHEMATICAL MODELING

Mass Conservation Equations

Energy Conservation Equations

NUMERICAL SIMULATION

RESULTS AND DISCUSSION

Effects of the Operating Parameters '.

Comparison with Existing Models

NOTATION

LITERATURE CITED

APPENDIX A. DERIVATION OF EQN. (2-a)

APPENDIX B. DERIVATION OF EQN. (15)

APPENDIX C. DERIVATION OF EQN. (27-a)




-97

-98

-99

1-1

1-1

1-2

1-7

1-14

1-19

1-20

1-22

1-26

1-29

1-30

1-31

1-33

1-35

1-36

1-38

CHAPTER IV CONCLUSIONS AND RECOMMENDATIONS IV-1

LISTS OF TABLES

Page


Table 1. Classification of fludized-bed dryers

based on operating modes 11-103

Table 2. Classification of the dryer types based on

structural and mechanical features 11-104

Table 3. Evaluation of the parametres used in the

semicompartment model 11-105

CHAPTER III MODELING AND SIMULATION OF A CON-

TINUOUS FLUIDIZED-BED DRYER

Table 1. Performance characteristics of the dryer

under various T. .111-39


under various Uq. .111-39


under various Tw 111-40


under various x Q 111-40


under various t ....-. 111-41

Table 6. Comparison of performance characteristic of

the dryer under adiabatic condition with

those under bed-wall heating condition 111-41

Table 7. Comparison of the performance characteristics

of the dryer based on present model with

those based on Palancz's model 111-42

ii

LISTS OF FIGURES

Page


Fig. 1. Single-stage cylindrical f luidiz ed-b ed dry er .. 11-106

Fig. 2. Schematic of a tapered fluidized bed dry er .... 11-107

Fig. 3. Two-stage f 1 u i d i z e d-b e d drying system 11-108

Fig. 4. Continuous multi-stage fl u idiz ed-bed dry er .... 11-109

Fig. 5. Schematic of centrifugal fl uidi

z

ed-bed

apparatus 11-110

Fig. 6. Vibro-f lui di z ed bed dryer 11-111

Fig. 7. Structural representation of a tapered

fluidized -bed dryer 11-112

Fig. 8. Schematic of a section of a centrifugal

fluidized bed 11-113

Fig. 9. Schematic representation of two-phase theory

of fluidization 11-114

Fig. 10. The variation with gas velocity of the

bed-to-surface heat transfer coefficient

without radiation transfer 11-115

Fig. 11. Typical r a t e-of -dry ing curve 11-116

Fig. 12. Schematic representation of three-phase and

semi-compartment model for a batch fluidized-

bed dryer 11-117

Fig. 13. Moisture transfer between phases with respect

to ith compartment 11-118

Fig. 14. Representation of compartments of upflow

emulsion gas 11-119

Fig. 15. Schematic representation of bubble-cloud

n,odel 11-120

Fig. 16. Moisture transport in a single bubble, its

surrounding cloud and in emulsion phase 11-121

Fig. 17. Representation of the exchange zone in cl oud .. 11-121

Fig. 18. Moisture transfer between phases 11-122

CHAPTER III MODELING AND SIMULATION OF A CON-

TINOUS FLUIDIZED-BED DRYER

Fig. 1. Schematic diagram of continuous drying in a

fluidized bed 111-43

Fig.. 2. Mass and energy transfer between bubbles and

emulsion gas 111-44

Fig. 3. Mass and enery transfer between solid

particles and emulsion gas .111-45

Fig. 4. Enery balance around the stagnant film

surrounding a solid particle 111-46

Fig. 5. Effect of the inlet-gas temperature 111-47

Fig. 6. Effect of superficial gas velocity 111-48

Fig. 7. Plot of length of the constant rate drying

period against the superficial gas v e 1 oc i t y . . . 1 1 1-49

Fig. 8. Effect of the dryer wall temperature 111-50

Fig. 9. Effect of the inlet-gas moisture content 111-51

Fig. 10. Effect of the mean residence time of

particles( . .111-52

lv

Fig. 11 . Comparison between adiabatic and bed -wall

beating cases 111-53

Fig. 12. Comparison of the present model with

Palancz's model 1 11-54

ACKNOWLEGEMENTS

I wish to acknowledge my co-advisers, Dr. F. S. Lai and Dr. L. T.

Fan, for their invaluable guidance throughout the course of this work.

Their enthusiasm and helpful criticism have been a constant source

of inspiration. I also wish to acknowledge the financial support

provided by the U.S. Grain Marketing Research Laboratory.

vi

CHAPTER I

INTRODUCTION

1-1

Fluidized-bed drying is one of the modern methods for drying.

In comparison with the conventional packed-bed or moving-bed dryers,

fluidized-bed dryers possess the following significant features:

1. Drying gas is locally mixed intensively during its passage

through the bed; consequently, the rate of mass and heat

transfer between gas and solids are high.

2. This extremely rapid heat transfer enables relatively high

inlet gas temperature to be used.

3. The time of drying is relatively short.

Because of its numerous advantages, fluidized-bed drying has been

increasingly applied in diverse industries in either the batch or

continuous mode. While batch fluidized bed dryers are often used

when the production scale is small or products are heat sensitive,

continuous fluidized-bed dryers are extensively used in processes

closely integrated with continuous production without intermediate

storage. Besides, the cost of drying per unit mass of product is

relatively smal 1

.

Although the theory of fluidization has developed rapidly in the

last two decades, this development is not amply reflected in the

study and practice of fluidized-bed drying. Conventional models for

a fluidized-bed dryer are mainly based on the overall mass and ener-

gy balances around the entire dryer. In addition to these mass and

energy balances, the models for a continuous fluidized-bed dryer co-

mmonly impose assumptions that

1. The bed temperature is uniform.

2. The outlet streams are in thermal and concentration equili-

brium.

1-2

3. The fluid-mechanistic behavior of the drying gas is homoge-

neous; in other words, the drying gas is not partitioned

into different phases of the fluidized bed, such as emulsion

and bubble phases.

Hence, all these models involve postulates which are not adequately

representative of the complicated phenomena occuring in a fluidized

bed dryer. This inevitably limits their range of applicability.

The overall objective of this work is to derive a fairly rigorous

mechanistic model for a continuous f luidized-bed dryer based on the

information generated through an extensive and critical review of the

available literature on this and the related subjects. In addition

to this introductory chapter, this thesis contains three other

chapters. An extensive and expositional review of the f luidized-bed

drying is presented in Chapter II. It covers the classification of

fluidized-bed dryers, phenomena of f luid iza t ion , transport processes

in a fluidized-bed dryer, drying characteristics of the solids, and

modeling of fluidized-bed dryers. Chapter III is concerned with

modeling and simulation of a continuous fluidized-bed dryer. The

model. based on the two-phase theory of f luidiza t ion , delineates the

intricate transport processes in the dryer. The proposed model, in

essence, is a significant amendment and extension of a comprehensive

mechanistic model which is available for a continuous fluidized-bed

dryer. Conclusions and recommendations are given in the final

chapt er

.

CHAPTER II

LITERATURE REVIEW

II-l

INTRODUCTION

This chapter presents an extensive and expositional review of the

published works on the various aspects of the f luidized-bed drying. The

review covers the classification of f luidized-bed dryers, phenomena of

f luidization, transport processes in a f luidized-bed dryer, drying charac-

teristics of the solids and modeling of f luidized-bed dryers.

CLASSIFICATION OF FLUIDIZED-BED DRYERS

Fluidized bed dryers can be classified mainly according to the follow-

ing criteria (see Tables t and 2)

1. Operating Mode . Fluid) zed-bed dryers can be operated either in

batch or continuous fashion. The latter can be further classified based on

staging modes (single stage or multiple-stage). Both batch and continuous

operations are often subgrouped according to the types of devices empl oyed

in facilitating fluidization (for example, vibrating or rotating devices, or

it can be of conventional stationary fluidized bed without any externa]

devices )

.

2. Structur al and Mechanical Feature s. Fluidized-bed dryers can also be

classified on the basis of their geometry and bed depth. The bed can be of

constant or variable cross-sectional area in the axial direction, with

cross-section being rectangular or circular. The bed depth can be deep or

shal low

.

A fluidized bed can be formed by synthesizing through various combina-

tions of these two criteria according to varying objectives. An example can

be a continuous, multi-stage, deep cylindrical -tapered fluidized bed for

drying (Brit. Chem. Eng., 1961). In this review, we shall present several

n-a

examples of each category. For additional examples, readers are refered to

the references listed in Tables 1 and 2.

Single-stage dryers

The best known dryers in this category are cylindrical ones (Fig. 1).

Their main advantages are simplicity in construction and operation, ease in

maintenance. low cost and the possibility of complete automation. Among

disadvantages must be included the relatively large unevenness of drying in

comparison with other types of f luidized-bed dryers, and convection of fine

particles of the dried material. To increase the uniformity of drying, some

dryers are built in a slightly conical form, with a tapered angle of 30-40°

(Fig. 2). The ratio of the cross section diameter of the upper part of the

chamber to the lower varies within the range of 10:1 to 50:3.

Multi-stage dryers

Multi-stage dryers are used mainly where, due to the sensitivity of the

dried material to relatively long exposure to elevated temperatures, the

drying temperature must be low. and. al the same time, where a final low

moisture content in the material is required. A two-stage dryer with back-

mixing/plug-flow bed (Fig. 3) is suitable for products that initially

release their moisture readily and later have a drying curve of decreasing

drying rate. These stages are so arranged that the product flow in the

counter-current mode to the drying gases, thereby reducing considerably

space requirements and construction costs as well as energy consumption.

Another example of multi-stage dryers is given in Fig. 4.

11 -3

Centrifugal-bed dryers

The centrifugal fluidized bed allows table, smooth fluidizalion for

large irregular particles of low bulk density at air velocities well above

those required for pneumatic transport. High velocities of air flowing in

the radial direction permit intensive heat transfer at relatively low

temperatures. Thus, this type of dryers (Fig. 5) can be used for initial

moisture reduction of sticky, high moisture and heat sensitive materials.

Vibrated-hed dryers

Some products to be dried have very wide particle size distributions

and others consist of rather large, oddly-shaped particles or agglomerates.

To dry such products, mechanical assistance is required to suspend the

particles in the drying gas. This can be accomplished by vibrating a

f luidized-bed (Fig. 6). A vibro- fluidized bed can be obtained in apparatus

of different construction by means of vibration of the drying chamber, the

bottom or baffles of the bed, and also by using special vibrating devices

inserted directly into the drying chamber. It has been shown (Slenov and

Mikhailov, 1972) that drying in the vibrated fluidized bed significantly

intensifies the drying process; the drying rate is increased several folds

with the effect of vibration; and sticking of the particles is avoided.

II-4

PHENOMENA OF FLUID1ZATION

When a fluid flows upward through a bed of solid particles at a rela-

tively low flow rate, the particles remain immobile or fixed. As the flow

rate of the fluid increases, the drag force acting on the particles will

also increase until a point is eventually reached where the drag force is

equal to the gravitational force holding the particles within the bed. At

this point. the particles will begin to move apart, becoming suspended in

the flowing fluid, and the bed will expand upwards. Under this condition,

the frictional force between a particle and fluid counterbalances the effec-

tive weight of the particle, and the bed is considered to be in the state of

minimum or incipient f luidization . If the flow rate of the fluid is in-

creased further, the movement of particles is intensified, and bubbling and

channeling of the fluid are observed. Such a bed is called a dense-phase

fluidized bed, bubbling fluidized bed or, simply, fluidized bed. A still

higher fluid velocity results in the solid particles being transported out

of the bed with the flowing fluid. A bed under such a condition is termed a

lean phase fluidized bed or. simply, entrained bed. Recycling of these

carried-away particles by means of a mechanical device, e.g., a cyclone, to

the bottom of the bed will give rise to the so-called "fast f luidizat ion"

regime, characterized by a high degree of particle turbulence. This is true

especially for the gas-solid system with fine particles since their terminal

velocity is relatively low.

For a liquid solid system with particles of moderate size and density

or gas-solid system with fine and light particles, an increase in the flow

rate of fluid above the minimum f luidization velocity tends to cause the bed

to expand smoothly without the formation of bubbles. Such a phenomenon is

called bubbleless or particulate f luidization . Nevertheless, liquid bubbles

II-5

can form if the solid particles are large and heavy (e.g., relatively large

tungsten particles fluidized by water).

Bubbles in a bubbling bed have been observed to coalesce and grow in

size. If the bubble diameter approaches to that of the bed, it will be in

the slugging regime; however, this topic is outside the scope of the present

review.

A fluidized bed system has numerous advantages over the fixed or moving

bed system. These include intensive or rapid local mixing of solids, th-

ereby inducing a uniform bed condition and high rates of heat or mass

transfer between the particles and fluid or between the wall and the bed.

On the other hand, the fluidized bed system has some undesirable

characteristics. For example, the extensive macroscopic mixing of solids

leads to their nonuniform residence time distribution; this tends to lower

the quality of the solid product, and also reduces the overall potential for

heat transfer, mass transport or chemical reaction involving the solids as

reactants. Furthermore, the existence of bubbles renders gas-solid contact

inefficient

.

Extensive research has been undertaken to facilitate our understanding

of the behavior of fluidized beds. The volume of publications on this and

related subjects is enormous. To survey comprehensively these publications

is almost an impossible task; thus, only those relevent to the present work,

which deals exclusively with gas-solid systems, are reviewed.

Pressure Drop and Minimum Fluidizing Velocity

Columnar bed . Fluidization is initiated when the pressure drop through

the bed becomes equal to the total effective weight of solids. In other

words (see, e.g., Kunni and Levenspiel, 1969),

11-6

(-AP)mf s f mf (1)

The superficial velocity of fluid at minimum fluidization, U „, appearsmf

in numerous correlations relating various variables and parameters defining

the state of fluidization; therefore, its accurate estimation or prediction

is important. While numerous methods have been proposed for estimating Umf

the most widely employed one is that obtained by equating the pressure drop

expression for the fluidized bed to that for the fixed bed proposed by Krgun

(1952); it is

WV edP 1.75

s mf

12

f 150( 1-C Jmf

.3.3s mf

(2)

where

ReSW

(3)mf n

If information on shape factor.<f> , and the incipient fluidized bed voidage,

e ,, is not available, eqn. (2) can be approximated by (Wen and Vu , 1966)mi

Remf

2Pf(P

s~P f)B(i

p 1/2[(33.7) 4 0.0408 — 2

1 £] ' - 33.7 (4)

Tapered bed. As stated at the outset of this section, under the condi-

tion of incipient f luidization , the total forces exerted on a solid particle

by the fluidizing medium equals its total effective weight. For a columnar

bed, the cross-sectional area is uniform along the axial direction. If it.

is assumed that packed bed of solids prior to the onset of fluidization is

essentially uniform, we can consider the balance between the overal frac-

tional and gravitational forces exerted on the entire bed in determining the

minimum fluidization condition. For a tapered bed, however, the pressure

II-7

drop through a given increment varies along the axial direction in which the

cross-sectiona] area of the bed increases. Thus, the overall pressure drop

through the entire bed height should be evaluated by applying eqn . (2) to a

section of the bed with a differentia] height and integrating the resultant

equation from the bottom to the top of the bed (Fig. 7). The final expres-

sion takes the form (Shi et a2- , 1984)

(Ap) - <(> U h Inmf loo

r H+hc

hf>„U

2hr

2 o

H

H + h( 5

)

where

and

150(l-e ,)mi

<VP»

(6)

1 .75(1-6

mf(*sV

(7)

It has been shown that the above expression reduces to that for the columnar

bed when the apex angle becomes negligibly small. Correspondingly, the

minimum fluidization velocity is evaluated as

<f>, U -J WH + *„U „J h Win1 mf o

T2 mf o o

f H + ho

h

0.5(WBf )(P8

-pf)ioWH

f H + 2hc

h(8)

Cen trifugal bed . In a centrifugal fluidized bed, particles are

fluidized in the centrifugal field. The bed usually consists of a cylindri-

cal basket which rotates about its axis of symmetry. The rotation of the

axis causes the particles in the basket to form an annular region at the

circumference of the basket. Fluid is injected inward through the porous

surface of the basket wall. When the rotational speed of the basket is

fixed, the pressure drop through a centrifugal packed bed increases as the

fluid velocity increases, and at a certain velocity, the particle at the

free board surface of the bed will begin to fluidlze. At this instance, the

pressure drop through the bed reaches a maximum and corresponding fluid

velocity reaches a critical fluidization velocity.

Fan et aj.. (1985) have proposed a mechanistic model for determining the

incipient fluidization conditions in a centrifugal fluidized bed. Unlike

many other models (see. e.g., Levy cl al., 1978; Kroger et uA. , 1979) it

takes into account the centrifugal acceleration and the curvature effect of

the cylindrical fluidized bed. According to this model, the pressure drop

through a packed bed in the centrifugal field with a differential thickness

of dr can be expressed in terms of the sum of the lirag force as correlated

by Ergun (1952) and the centrifugal and dynamic force experienced by it

(Fig. 8). The maximum pressure drop through this confined volume element is

evaluated by equating the product of the pressure drop through it and the

radial area with the effective weight of the particles contained in it.

Integrating over the radius of the bed yields the overall maximum pressure

drop across the entire bed. It has been derived as

r

(-AP) =<f> U r In — - d> U

2r2 (— -)

max 1 oc o r. 2 oc o r. rl i o

2 2 2

+

Pf" . 2 2, ffVVl 1,

* ~2- (ro

ri> '

1"<1 I) (9)r . r

i o

The corresponding critical fluidizing velocity, JJ is evaluated through

II-9

r.! 2, 2,1 1,, 2

[Vo lnr~ < Vo'r" - r )]U

oc+ Vo (Vr

i>Uoc

(10)

The estimated minimum fluidization velocities are In good agreement with the

experimental data.

Vibrated bed. The vibrated fluidized bed consists essentially of a

vibrating screen through which air is forced from a plenum beneath it.

Suppose that the horizontal screen is subject to vibrations of amplitude a

and angular frequency o. If the vibrational acceleration oxj is less than

that due to gravity g, the solids will remain in contact with the screen.

2However as vibrational intensity increases (ecu /g>l ) a point will be reached

where the bed begins to detach itself from the screen and move freely under

the influence of gravity. This state is defined as that of incipient

vibrof luidization. The incipient vibrof luidizat ion velocity U can bemv

significantly lower than the minimum fluidizing velocity (U ) of themf s

corresponding static bed. A theoretical expression for U /(U „) has beenmv mf s

derived by means of force and momentum balances (Jinescu. 1971)

mv ,1+k, ,oro,

<v>;(^' '^ (n)

In this expression, k is a collision elasticity factor which must be deter-

mined experimentally. It approaches zero for fine particles and unity for

coarse particles. The factor j. which is a small integer, is given by

, adJ "

1(12)

P

n-io

where the sum (t + t ) is the total flight time, t being the time of

ascent of the bed and t. the time of descent; t is the period of thed p

vibrations. The pressure drop under the incipient fluidization condition

with vibration is interrelated to that without vibration as follows:

i

T^JL m l

„. 935 (

V34« (Syf ,0.606 ^ 3.837(13)

1 mf'sHf g s

Gupta and Mujumda (1980) have proposed that for efficient operation in

a vibrated fluidized bed, the air velocity must exceed a "minimum mixing

velocity", U . at which solids mixing could be observed visually. Theymm

have contented that thi s value is more practical from the design standpoint,

than the one determined from the pressure-drop-velocity curve. Based on

their data for spherical and near spherical particles, the following cor-

relation has been proposed for this quantity

TjJfi- . 1.952 0.275 (2y!) - 0.686 (^!) 2(14)

where (U _) is the theoretically computed minimum fluidization velocity of

the bed under the non -vibrating condition.

Bed Expansion

Beyond the point of incipient fluidization, further expansion of a

fluidized bed is related in a complex manner to numerous parameters such as

the physical properties of solids and fluids, the gas flow in excess of the

minimum fluidization velocity, particle size and size distribution, and the

bed-height to diameter ratio. Superimposed on this complicated relationship

is the nonumiform voidage in the bed as identified by three distinct zones:

(1) a distributor effect zone (2) a zone of constant bed density, followed

11-11

by (3) a zone of continuously decreasing solids density, which makes it

difficult to estimate fluidized bed heights. Furthermore, because of agita-

tion of solids by the fluidizing medium, the top surface of the bed is

usually uneven and oscillating. Thus the voidage, €, and the bed height,

H_, can only be considered as time average values.

The published work on describing bed expansion on fluidization is

general ly based on ( 1 } the two -phase theory of fluidization, which assumes

that the gas in excess of that required for minimum fluidization velocity

will pass through the bed in the form of bubbles (2) two-phase theory and

bubble properties

.

The relationship between voidage, 6, and fluidizing velocity, U , haso

been commonly used to determine particulate phase expansion. It has been

known (see, e.g., Steinour, 1944; Lewis et al_., 1949; Lewis and Bowerman

,

1952; Richardson and Zaki , 1954) that for a particulate or ideal fluidized

bed, a linear correlation on a logrithmic scale exists between bed voidage

and fluidizing velocity. It takes the form

U = eV (15)o t

where n is a function of d/0 and Re (Richardson and Zaki , 1954; Richardson,

1971). This equation can also be expressed in terms of the linear fluid

velocity, U /f, aso

Uo n-l„~ = 6 U

t(16)

According to Kwauk (1963), the above relationship can be generalized in

terms of the relative linear velocity between the fluid and particles as

follows

:

V up

€ l-e

11-12

(17)

It is worth pointing out that eqn . {16) is also valid for nonideal or bub-

bling fluidized bed (Richardson. 1971). Accordingly. the same can be

infered for eqn. (17).

If the so-called two phase (bubble-emulsion) model of fluidization is

adopted (see Fig. 9), e can be related to the corresponding parameters in

the model as

6 = ib

+ (1 V emf " 8I

l-e -- {1-6 )()-e ) (19)b mi

Equation (2-17) can then be rewritten Tor gas-so] id system as

u u

[«K+(i-«Je .]n"V (20)6+11-6 )« (l-«.)(l-6 ,)

Lb

v

b' mfJ

tb b mf b mf

Many empirical correlations have also been proposed for estimating

fluidized bed expansion for a gas-solid fluidized system (see, e.g.. Leva,

1957; Lewis, 1949; Bakker and Heertjes, 1960: Shen and Johnstone, 1955). A

correlation based on the regression analysis of the available data has been

proposed by Babu et aj. (1978). It is

H. 14.311 (IMIF)°- 738d

'"%»"«f , mf p s

H 0.937 0. 126 ' '

mf I] pmf g

This correlation is independent of the effect of column diameter for D >c

0.065 (m). It is, therefore, relatively reliable for scale-up. Comparison

of the calculated and measured expansion ratio based on the correlation

reveals that about 90% of the data Is predicted within 1 12*.

11-13

Fluid Mechanistical Structure of the Bed

Formation of phases . It is now generally accepted that a one-phase

representation of a fluldlzed bed (pseudo-homogeneous model) is inadequate

for a bubbling fluidized bed, and that at least two hydrodynamically dis-

tinct phases should be visualized. One is the phase where gas percolates

through, rather as in a packed bed, and the other the phase where much of

the gas, existing in the form of bubbles, is out of contact with solids. In

fluidization. the former is called the dense or emulsion phase and the

latter the lean or bubble phase. Based on this visualization, a well-known,

two-phase theory of fluidization has been developed (Twoomey and Jonestone,

1952). The theory further assumes that the flow of fluid in excess of

minimum fluidization passes through the bed as bubbles and that the emulsion

phase is similar to the bed at minimum fluidization. While various models

have been proposed for fluidization, most of them are based on the two-phase

theory. However, these models differ substantially regarding the assump-

tions of flow of gas through the two phase, the extent of mixing, and the

mode of interphase exchange (see, e.g., Van Deemter, 3961; Orcutt et al-.

1962; Davidson and Harrison. 1963; Partridge and Rowe , 1966; Kobayashi et

al. , 1967; Kato and Wen, 1969).

Another representation of the fluidized bed is the three-phase model

(see. e.g., Kinni and Levenspiel, 1969; Frey and Potter, 1976; Mao and

Potter, 1984) which considers the cloud and wake region as a separate phase

(the cloud wake phase) in addition to the emulsion and bubble phases.

Numerous experimental envidence exists to indicate that characteristics

of a fluidized bed depend heavily on the behavior of bubbles. It affects

the relative magnitudes of the bubble and emulsion phases, the mixing of

solids and that of fluid, the interactions between the solids and fluid, and

11-14

those between the contents and the wall of the bed. It has been recognized

that the coalescence of bubbles leads to an increase in the bubble size in

the direction of flow. Various correlations have been published for

predicting the axial distribution of bubble sizes; the most recent ones

include those by Mori and Wen (1975), Rowe (1976), and Darton et al. (1977).

Among them, the mechanistic model proposed by Darton et aj.. (1977) and the

semi-empirical equation by Mori and Wen (1975) appear to be more accurate

than the others in predicting the bubble size; therefore, they have been

recommended for use by different researchers (see, e.g., Grace, 1981; Yates,

1983)

.

Almost all the available correlations suggest, that the bubble size

increases indefinitely in the direction of flow; however, it has been ex-

perimentally determined that the bubble size reaches a maximum in a

fluidized bed containing fine particles (Geldart, 1973; Davidson et al.

.

1977). Furthermore, the growth of bubbles will be restricted in a bed

equipped with internals. All of these indicate the difficulty of estimating

the bubble size in a fluidized bed. Thus, various researchers have assumed

the bubble size to be constant in modelling a fluidized bed; that is. no

coalescence exists between the bubbles (see, e.g., Hoebink and Rietema,

1980(a) and 1980(b), palancz, 1983). Another approach recognizes the growth

of bubbles, but employs the effective bubble size throughout the bed or

simply treats it as a parameter (see. e.g., Kunni and l.evenspiel , 1969).

Although a substantial number of correlations have been proposed for

the bubble size velocity, the semi - empi rica 1 equation proposed by Davidson

and Harrison (1963) is overwhelmingly popular; it is

Ub " (V°-f>

+ 0-71K8V- 5

< 22 »

This expression has been shown experimentally to give a good estimation of

the bubble rise velocity (Kowe and Partridge, 1965, Kunni and Levenspiel,

1969)

.

Gas movement and gas mixi ng. The pioneering work by Davidson and

Harrison (1963) on the characteristic features of a fluidized bed in the

light of fluid mechanics has paved the way for the later systematic studies

on the subject. The model suggested by Davidson and Harrison inciudes some

bold assumptions: (1) the original two-phase theory of Twoomey and Johnstone

(1952) is valid and (2) the bubbles are circular in shape and behaves as if

they were in inviscid liquid. Although the model is simple, it correctly

predicts some significant flow characteristics, including the flow pattern

of gas and movement of solids in the vicinity of bubbles, pressure distribu-

tion, and the formation of cloud verified experimentally by Rowe (1964).

The assumption of circular bubbles is obviously an over-simplified one.

Experimental evidences show that a bubble normally has an indented base of

solid particles, which is defined as the wake region. A theory based on

this more realistic picture of bubbles has been proposed by Murray (1965).

A variety of models have been proposed for the mixing of gas in the

emulsion and bubble phases. These include, among others, the plug flow,

perfect mixing and dispersive flow models. The flow of gas bubbles is

usually assumed to be in the plug flow mode, that is, individual bubbles

rise at a uniform velocity without coalescence. This is obviously an over-

simplified assumption. Nevertheless, it is widely employed and yields

surprisingly good predictions. No definitive experimental evidence is

available to base the assumption about the flow pattern through the emulsion

phase; thus, widely varied flow patterns are assumed in different models.

They include the perfect mixing, plug flow, tanks in-series, dispersive

flow, stagnant and down flow.

If the performance of a f luidized-bed dryer is controlled by diffusion

of moisture through the solids, it will be relatively insensitive to the

flow patterns of gas. Moreover, if most of the gas passes through the bed

as bubbles, i.e., U >>U ., it is obvious that the contribution of the emul-o mf

sion gas to the performance of the bed will be negligible, and the selection

of the flow pattern of emulsion gas will not be critical. Except for these

two special cases, the performance of the bed will bo strongly influenced by

the flow pattern of gas.

Solids Movement and Mixing

Experimental observations of the movement of solid particles in a

fluidized bed have revealed that the solid particles move in a fairly random

manner, and the movement and mixing are induced by the rising bubbles {Rowe

and Partridge, 1965; Kunni and Levenspiel, 1969; Cranfield, 1978; Shi and

Fan, 1982). Although the theory of Davidson and Harrison (1963) has been

successfully employed in explaining the local movement of particles around a

bubble, apparently it is incapable of explaining the global movement of

solids and the resultant solids mixing in a fluidized bed. Three major

mechanisms of the particle movement causing this mixing have been reported

(Rowe and Partridge. 1965; Cranfield, 1978). They include; (1) eddy diffu-

sion, (2) bubble-wake induced drift, and (3) bubble-induced drift. The

mixing mechanism could be one, two or all three, depending mainly on par-

ticle size.

An equation, analogous to Kick's equation, has been commonly used to

describe the solids mixing. Its expression is

11-17

3C a2c , „ 3C

5T="sa —i + "5 to <* D

sl §T> < 23 >

OZ X

where a = 0, 1, representing cylindrical and sperical geometry of the bed,

respectively.

To facilitate the determination of the axial dispersion coefficient,

I) , and the lateral (or radial) dispersion coefficient, D , experiments

are usually designed in such a way that these coefficients can be found

separately from the experimentally determined solids concentration, C .

s

Note that D and D are sometimes named diffusion coefficients. Based onsa se

the experimental results, a substantial number of correlating equations have

been proposed to determine the influences of significant parameters, such as

the min j nuim f luidi zation condition, fluid vel oci ty , bubble size , particle

properties and bed geometry. on the magnitudes of the dispersion

coefficients

.

Solids motion in the vertical direction is mainly induced by the motion

of bubbles; thus, it is somewhat inappropriate to consider a of diffusion

process of the Fick's law type for vertical solids mixing. Arastoopour and

Gidaspow (1979) have proposed a model for vertical countercurrent solids gas

flow in fluidized bed reactors. According to their model, vertical counter-

current flow of gas and solids of a uniform size can be mathematically

described by means of one-dimensional, isothermal steady state mass and

momentum balances. it is suggested that from hydrodynamics in the fluidized

bed, the empirical transfer or diffusion coefficient approach be replaced

with mixing predicted from the laws of motion of the gas and the solid

particles

.

11-18

The time constant for drying solids is usually much longer than that

for the mixing or the solids. Thus an assumption that the solids in a free-

bubbling bed is uniformly mixed is reasonable. Actually, this assumption

has been confirmed by the experimental work by Vanecek et al. (1970), Brauer

et aj_. (1970) and Hoebink (1977).

11-19

TRANSPORT PROCFSS

Gas' Interphase Exchange

Recognizing the existence of bubble and emujsion phases in the

fluidized bed, questions arise as to the mode and rate of transfer of a

fluid component between these two phases and as to the manner for expressing

them in terms of measurable parameters. The first approach is to correlate

the interphase exchange coefficient, defined empirically, similar to the

mass transfer coefficient (see, e.g.. May, 1959; Van Deemter, 1961). The

second approach is to relate it to the characteristics of the fluid in

general and bubbling mechanism in particular (see, e.g.. Partridge and Rowe,

1966; Kunni and I.evenspiel, 1969; Davidson et al . , 1977). The latter is

based on the fundamental mechanism, and therefore, it has the advantage that

we can use it with confidence in scale-up; this is not usually the case for

the former which is purely empirical.

Various expressions for the interphase exchange coefficient have been

published. One of the common assumptions is that the gas enters from the

bottom part of the bubble and leaves at the top (Partridge and Rowe, 1966

and Kunni and Levenspiel 1969); this generates a closed circulation of gas

within the so-called cloud region. The interphase transfer of gas is

usually considered to include two steps in series (see, e.g., Kunni and

I.evenspiel, 1969). One is expressed in terms of the exchange of gas between

the bubble and cloud-wake phases, involving both convective and diffusive

transfer, and the other between the cloud-wake and emulsion phases, involv

ing mainly the diffusive transport. Accordingly, the overall interchange

coefficient for mass transfer can be expressed as (Kunni and Levenspiel,

1969)

1= _J_ _J_

K. " K.+

K(24)

be be ce

The analogous form for heat transfer is

1 1 1

ST ' s-4

s- (25)

be be cc

The subscripts h, c, e represent bubble, cloud and emulsion phase

respectively. It is worth pointing out that all the transfer coefficients

in eqns. (24) and (25) should have the same volume base (which could be

either bubble, emulsion or total bed). Thus (K, ), , (H, ), etc. wouldbe b be b

denote the bubble volume on which transfer coefficients are defined.

Partridge and Rowe (1966) have assumed that the bubble and cloud-wake phases

are perfectly mixed; thus the limiting step of the mass transfer is between

the cloud-wake and emulsion phases; in other words, Kt -> oo and H, -*• <o inbe be

eqns. (24) and (25), respectively.

Bokur et al . (1974) have reported that when U > 2U „ the cloud suro mf

rounding a rising bubble is very thin, and therefore, interchange can be

considered to occur only between the bubble and emulsion phases. Davidson

et a\. (1977) has shown that the cloud is not closed in the region below the

bubble and have chosen to ignore the recirculation of gas in the cloud

region. In addition, several empirical correlations have been proposed.

Among them, the equations proposed by Kobayashi et al. (1967) and Grace

(1981), which incorporates the effect of bubble interaction, appear to

represent the experimental data reasonably well.

11-21

Gas-to-Particle Heat Transfer

Mechanism . The flow of gas around particles in a fluidized bed is

essentially streamline and the local Reynolds number is correspondingly

small; thus, particle-to-fluid heat transfer coefficients are normally quite

low. Typical values cited for these coefficients are 6-23 kW/m k

(Botterill, 1975). On the other hand, the size of the bed particles is

small and their number high, which leads to a large total surface area of

solids per unit volume of the bed exposed to the flowing gas; values ranging

2 3between 5000 to 45000 m /m are cited (see, e.g., Botterill, 1975). As a

result , the overa 11 heat transfer rate between the particles and the main

stream of gas is very rapid

.

There are a number of factors which will affect the heat transfer

process between the particles and fluidizing gas. The presence of adjacent

particles affects the thickness of the gas-f i 1m surrounding the individual

particles while gas by-passing zones of the bed will adversely affect the

rates of the heat transfer. The extent of by-passing is dependent on the

bed material, degree of f luidizat ion , design of the apparatus and consequent

gross mixing patterns. A rigorous analysis of the heat transfer process

will, therefore, require a large number of parameters, which is not

feasible. Instead, empirical or semi -empirical correlations for overall

coefficients are often sought for design purpose.

Correlations for Overall Transfer Coefficients . Because of its impor-

tance, an enormous amount of research has been carried out on this subject

(Barker, 1965). Based on the available experimental data, numerous correla-

tions have been proposed. Nevertheless, the most striking feature of the

published expressions is the lack of agreement among observers with over a

thousand fold variation (Kunni and Levenspiel, 1969). It is, there-

fore, necessary that the experimental conditions are similar to those under

our consideration and great caution should be taken when extrapolating the

data. In particular, we must see what global flow pattern is assumed for

the gas (e.g., plug flow or back mixing) since different flow patterns in

terms of macromixing will yield substantially different heat transfer

results.

The analysis of the published works, have indicated (Kunni and

Levenspiel, 1969) that measurements based on a plug flow model present a

more consistant pattern than those based on a perfect gas mixing model. The

former can be correlated by the equations:

Nup

- 0.03 Rep3

(26)

This relationship, actually, is an overall correlation based on the works of

several investigators (Richardson and Ayers, 1959; Kettering et al., 1950;

Heertjes and McKibbens, 1956; Donnadieu, 1961; Walton et al. . 1952). It is

applicable approximately in the range

Re < 100 (27)

For the same plug flow model, a different correlation has been proposed

by Chang and Wen (1966). They measured the fluid-to-particle heat transfer

coefficients in a baffled fluidized bed of large Raynolds number under

transient conditions. The correlation based on their experimental results

takes the form

Re

Jh

= 0.097 (-|2 )

°' 5(Ar)

" 2(2R)

with

Re

e (29)

and

5.86x10 < Ar < 2.55xl07

(30)

Recently, microwave heating technique under unsteady-state conditions has

been employed to collect fluid- to- particle heat transfer data. This has

resulted in a relatively uniform temperature of the bed. consequently.

reducing errors due to the temperature gradient within the bed and hence

particle-to-particle heat transfer. The following correlation has been

proposed based on the experimental findings (Bhattacharyya ami Pei, 1974)

„ „... Ar ,0. 25J = 0.043[ -] (31,

(Re /e)P

with

0.02 < —— < 10 (32)

(Re /€)P

For additional information, readers are referred to Frantz, 1961; Ferron,

1962; Bradshaw. 1963; Barker. 1965; Gupta. 1974; Balakrishnan , 1975; McGaw,

1977; Selzer, 1977; and Botterill, 1981.

Bed-to Surface Heat Transfer

Mechanism. In numerous fluidizcd bed processes, it is necessary to

transfer heat between the bulk of the bed and a surface. The latter can be

the surface of an immersed cooling or heating coil through which the heat

transfer medium is circulated or it can be the wall of the column containing

the bed. The variation with gas velocity of the bed-to-surface heat trans-

fer coefficient can be characterized in three regions under the condition

that radiation transfer may be neglected (see Fig. 10): (1) the region

11-24

below U . where the bed is in the packed state and the heat transfer coeffi-mf

cient is low, (ii) the region between 11 _ and an optimum velocity where themr

coefficient increases sharply to a maximum value, (iii) the region above the

optimum velocity where a gradual decrease sets in. It is generally believed

that the sharp increase in the heat transfer coefficient above U „ is causedmf

by bubble-induced particle motion at the transfer surface while the decrease

above the optimum velocity is the result of restricted particle surface

contact under conditions of high bubble flow (Yate, 1981).

Jt is generally accepted that heat transfer to an immersed surface may

be considered to comprise three additive components (Botterill, 1975; Yates,

1983)

(i) the particle convective component accounting for the conduction

heat transfer across the gas layer separating the solid particles

and surface. It depends on the contact time and thermal

properties of both gas and solids,

(ii) the gas convective component which can be attributed to the

interstitial flow in the case of emulsion phase contact and the

bubble flow field in the case of bubble contact.

(iii) the radiative component that becomes important only at tempera-

tures in excess of approximately 600°C.

By combining these three components, an overall prediction for heat transfer

between the bed and an immersed surface can be achieved

.

Corre la tions for overall coef ficients . A correlation for the overall

coefficients of the heat transfer between the wall and the bed has recently

been developed (Bock, 1983), which incorporates the convective, conductive

and radiative conponents of the wall-to-bed heat transfer.

The correlation for the gas convective contribution is (Baskakov et

al., 1973)

h - 0.009 -S pr1/ '!

JKr (U/IJ )

' 3(33)

gc d maxP

with (J in the range

I) „ < II < U (34)mf max

The radiative component h is (Hock, 1983)R

T - ThR

. 0.04 * 0-[(Tw

- TB

)

ln/100]

3[l t ( T

W-

T

B)

2] (35)

where

T T

(TW -Vln "

J1"T-

J (36)

TB

The particle convective heat transfer coefficient is evaluated as (Bock,

1983)

1 - 6

V =

;c -g (37 >

h (1-e .) 2 (k p c )max mf p p p

where

4k d

hmax^ l(1 < F' 1 "' 1 +5i» - « < 3S >

P P

and

2B = 4X(^ 1) (39)

The mean free path of gas molecules, X, can be calculated from

. i6 rwr u

11-26

y in eqn. (39) is the accommodation coefficient, accounting for the incom-

plete energy transfer during a molecule-wall collision. It is determined

through the following relationship;

i«*10<i- 1) -0.6- (JSSUi)^ (41)

A

The constant Cft

has different values for different gases (see Boch , 1983).

For air.

CA

' 2-8 (42)

The contact time of the particle phase, t, in eqn. (37) is defined as

t = u-y/fb (43)

The constant C in eqn. (37) counts for the roughness of the particles and

the heat transfer surface. C = 3 was reported for all the tested particles

(Bock, 19B3). The correlations from the three components of heat transfer

are summed to yield the overall bed-to-surface heat transfer coefficient, h.

as

b. = h * h + h„ (44)gc pc Rx

'

The validity of the proposed correlation has been tested over a wide range

of operating conditions. The predictions of the model show good agreement

with many experimental observations.

Relatively simple correlations neglecting the radiant component, have

also been developed. They are mainly based on the film or penetration

theory. A review of the relevant correlations are available ( Saxena and

Gabor, 1981).

Most of the proposed correlations for the heat transfer coefficient in

gas-solid fluidized beds are for relatively small particles. Attempts to

extrapolate these correlations to large particles have been unsatisfactory

11-27

(see, e.g., Decker and Glicksman, 1983). For a small particle system,

bubble dynamics and bubble-induced motion of solid particles play a very

significant role in the sense that the predominant mechanism of heat trans-

fer is due to particle convection (Mathur and Saxena, 1985). Consequently,

the heat transfer coefficient, h, decreases rapidly as the average diameter

of the small particles increases. In a bed of large particles, the inter-

stitial gas velocity is much greater than the bubble velocity, and as the

fluidizing velocity increases this slow-moving bubble regime changes to the

rapidly-growing bubble regime and finally to the turbulent regime (Catipovic

et al . , 1978). It has been shown (liorodulya et al . , 1980) that in turbulent

flow regime, the solids mixing is poor. Under this circumstance, the heat

transfer is induced mainly by the convective flow of gas surrounding the

particles and to a lesser extent by a steady state heat conduction due to

contact between the particles and between the surface and particles. For

such a system, the convective contribution of gas is far more important than

the convection contribution of the particles. As a result, the overall heat

transfer is intensified with the increase in the particle diameter.

Various investigators (Butterill et al. , 1981; Ganzha, et al. , 1982;

Decker and Glicksman, 1983; Mathur and Saxena, 1985) have recently developed

correlations for bed-to-surface heat transfer in large particle fluidized

beds. In defining "small" and "large" particles quantitatively, two powder

classification schemes have been used. The classification scheme proposed

by Geldart (1973) is based on the hydrodynamic behavior of particles. For

particles of small size or low density, bed expansion occurs well before

bubbling commences; in other words, minimum bubbling velocity U is greatermb

than minimum fluidization velocity, U For particles of large size or

high density, the reverse is true. Since both minimum fludization and

11-28

minimum bubbling velocities can be related to the diameter and density of

particles, the criteria for the classification of particles is set according

to particles diameter and density. Geldart's groups B and D particles are

usually referred to as small and large particles respectively. His group B

particles are defined as particles with 40 f/m < d < 500 /im, and 1400 kg/m3

3< P

s< 4000 kg/m . His group I) particles, on the other hand, satisfy the

2 3criterion that (p -p )d > 10

s g P

Saxena and Ganzha (1984) have shown that while Geldart's group I) par

tides will exhibit the hydrodynamie bubbling behavior of a large particle

bed, the heat transfer phenomenon of the bed may still reflect that of a

small particle system. Consequently, a desirable particle classification

can only be achieved if the fluid flow and heat transfer behavior are con-

sidered simultaneously. Note that Nusselt number representing heat transfer

behavior and Reynolds number characterizing fluid flow conditions are re-

lated to each other through their common dependence on Archimedes number.

Based on this fact, they obtained a powder characterization scheme by con-

sidering the Archimedes number together with the Reynolds number at minimum

fluidization. It classifies powders into three different groups. Their so-

called "large" particles are from groups 11(B) and III, which are defined as

5 fi

1.3 x 10' < Ar < 1 .6 x 10 (45)

Ar > 1.6 X 106

(46)

respectively.

11-29

Based on numerous published expressions for the bed-to-surface heat

transfer coefficient, a general correlation based on Geldart's Powder groups

has been proposed {Butterill et aj_. , 1981), which gives

i „-„, 0.6. -0.36 0.2h - 35.8 K d p 141)max g P P

for the group H particles, and

k

h -jS 0.843 At015

pc.max dP

(18)

p 39h = -S 0.86 Ar

( 4 q)pc d C*-MP

h = h < h (50)max go pc, max > '

for the group I) particles.

For a large particle fluidized bed, the convective contribution of gas

to overall heat transfer coefficient is the most significant factor. Unlike

the conductive heat transfer, the heat transfer by convective flow of gas is

not much affected by the shape and roughness of the particles. This renders

the modeling of the heat transfer between the particles and a surface rela-

tively simple. A mechanistic model for heat transfer between a fluidized

bed of group 111 particles and an immersed surface has been proposed (Kanzha

et aA., 1982); it involves parameters that can be easily determined. The

model assumes that an orthorrombic configuration of particles are arranged

around the heat transfer surface. It. is further assumed that the particles

can be replaced by equivalent cylinders whose volumes are the same as those

of the particles and of a unit diametcr-to height ratio. All the resistance

to heat transfer is considered to be confined to the first row of the par-

ticles near the heat transfer surface. The heat is transfered by conduction

through the gas lens (with a diameter equal to that of the equivalent

1 1 -30

cylinder), between the surface and the particle. Their proposed expression

is

Nu = 8.95(1 f )

0667. 0.12Re°'

8Pr043

- iiî° '

'

^.„,

0.8 ,OJ '

e

Mathur and Saxena (1985) have developed a new correlation which is

based on a total of throe hundred and thirty-six data points and the known

mechanistic details of heat transfer by particle and gas convection. The

proposed correlation takes the form

Nu • 5.95 (t-E)2/3

. 0.055 Ar°

'

3Re°

'

2Pr

1/3(52)

it is accurate within t 35* for gas - fluidized bed systems characterized by

Ar > 130,000.

Models with mure involved mathematical treatment can be found in the

work by Adams (1982), and Deck and Glicksman (1983).

Partlcle-to-nas Mass Transfer

Mechanism. The overall Particle- to- Gas mass transfer generally is

governed by (i) the density or the fluid; (ii) shape, size, and density of

the solid particles; (iii) diffusion coefficient of the material being

transferred; (iv) geometry of the system; and (v) operating conditions such

as the flow rate of the fluid, bed height, voidage of the bed, the bubbling

behavior and its accompanying features of gas by passing and channelling.

In flutdized-bed drying, particle-to-gas mass transfer is caused by

vaporization of moisture at. the interface and its successive migration into

the bulk phase of the drying gas. The vaporization of moisture at the

interface depends on the interface temperature, gas moisture content, etc.

The heat transfer across the interface resulting from a difference in tem-

perature in two phases will raise the interphase temperature relative to the

11-31

bulk-phase temperature, and the equilibrium at the Interface will then vary.

Consequently, the particle togas mass transfer in fluid! zed-bed drying is

related to the corresponding heat transfer process. Rut in practical ap-

plications, in order to employ the fluidization characteristics to the

maximum extent possible, drying operations in a fluidized bed is often

conducted in such a way that, the mass transfer resistance at the interface

between the gas stream and the solids is predominant or, in other words,

drying operations mainly occur in so-called constant-rate drying period.

Under this circumstance, the interface temperature, and the equilibrium at.

the interface remain unchanged if the same thing can be said for the state

of the drying gas. As a result, the mass transfer coefficient together with

concentration difference can appproximafe the mass transfer rate with

reasonable accuracy. This is especially true for the continuous operations

since the emulsion gas is commonly assumed to be completely mixed with

constant temperature and moisture content.

Correlation for overall mass transfer coefficients . Considerable atten-

tion has been paid to the gas-solid mass transfer and numerous attempts have

been made to derive analytical expressions for the transfer process.

Because of the large number of variables involved and inadequate knowledge

of the flow mechanism in the bed, purely mathematical or theoretical con-

siderations are not much favored in expressing the relation between

variables and more often than not, resort is taken to empirical correlations

developed from the experimental findings.

The conventional method for relating the mass transfer coefficient,

operating conditions, and the physical properties of the fluid is through

the dimensionless groups. Attempts have been made to correlate the ex-

perimental results in terms of the mass transfer factor (J ) or Sherwood

11-32

number (Sh) to the various forms of Reynolds number. To incorporate the

effect of the bed expansion, the void fraction has also been used to improve

the correlation.

Based on the available experimental data for the various systems

reported in the literature, a generalized correlation has been developed

(I)wivedi and Upadhyay, 1977 ) . which can bo used for the design of a fixed-

bed or fluidized bed process unit. It is

0.765 0.365

d *„ 0.82 0.386 (53)Re He

with

Re > 10(54)

and an average standard deviation of 17.95% with the experimental data. The

proposed correlation resulted from iterative least square analysis with

minimization of residual errors, hence is purely empirical in nature.

Starting from a theoretically derived relationship.

Sh oc f(Re*). (5S)

Paneey et aj.. (1981) have obtained, by regression analysis, the following

correlation for large particle fluidized bed system

eSh - ().95Re Sc^ (53)

which has been shown to be an approximation of the Nelson-Gal loway-Rowe

asymptotic expression (Nelson and Calloway, 1975; Rowe, 1975).

A correlation based on the data obtained in both gas- and liquid-

fluidized systems has been proposed (Reek, 1971) in the following form

(57)st sc

2/3. -g£ eSc

2/3. (o.e * o.i)(-2V -"

for

11-33

U d

50 < -2-£ < 2Q00

0.6 < Sc < 2000

(58)

(59)

and

0.43 < 6 < 0.75( 60 )

According to Beek (1971), the expression seems to be the most accurate and

reliable representation of available data within the applicable range of

parameters

.

Since the convective transfer coefficients for heat (h ) and mass (K )

P Pg

are both dependent on the How of the air, attempts have been made to corre-

late these two coefficients. This is especially important in the sense that

relatively few investigations have been made into the study of mass transfer

process in different systems and under various operating conditions. An

empirical relationship between heat and mass transfer coefficient has been

developed (Holman, 1972), which takes the form

h e 2/3

r = PP

CP

(F7' (en

PE

Attempts have also been made (Kato et a!, 1970) to adapt mass transfer

correlations obtained from fixed beds for fluidized beds by employing bubble

assemblage model. It has been shown that the following proposed mass trans-

fer correlations in fixed beds (Kato et yj , 1970)

Sh/Sc/3

- 0.7 2 [.<ep(d

p/H

f)

- 6!

- 95(62)

for

0.1 < Rep

(dp/H

f

)"- 6< 5 (63)

II -34

sh/sc1/3

. i.asfKepiyn/- 6!

- 63(64)

for

5 * % (W°"e

* '°3

(65)

can be applied to the bubble assemblage model to calculate the particle -

gas mass transfer coefficient in fluidized beds by

(i) using the gas velocity equivalent to U, . the bubbleb

velocity, in calculating the mass transfer coefficient

in the bubble phase

(ii) using the gas velocity equivalent to II „ in calculatingmf tJ

the mass transfer coefficient in the emulsion phase with I' to beb

calculated as follows:

o. « i.i u ,/e , (66)b mf mf |DD '

and

Ub

= 0.711 /edb (67 )

where

db

= l - 4 Vi-'r 11411!

1 (88)ml

(1.10 ,. /E\'1

mf mi

1 0.708p d (U/U )g (69)p p mf

Miscellaneous other correlations may be found in Upadhyay and Tripathi

(1975), which contains some 365 references. All of the above mentioned

research on mass transfer in fixed and fluidized bed was based on the as-

sumption that heterogeneous system can be treated at least in principle as

1 1 -35

quasi homogeneous systems. Also, the proposed mass transfer coefficients

are obtained based on steady state transport processes. Some recent works,

however, do consider the transient component of fluid-particle mass transfer

in fluidized-bed operations (see, e.g., Howebink and Rietema, 1980(a) and

1980(b)). Detailed discussion of their model is presented in section

MODELING OF FI.UIDIZED RED DRYING.

DRYING CHARACTERISTICS OF SOLIDS

The drying characteristics or a given species of solid under specified

conditions of the drying medium are required to set up drying schedules and

to determine the size of a dryer. These characteristics are often reflected

in the drying rate curve. A typical drying rate curve is represented in

Pig. 11. While different soiids and different conditions of drying often

give rise to curves of very different shapes, a drying rate curve usually

exhibits two major parts. the so-called constant and the failing rate

periods, as marked on the figure.

Constant Drying Rate Period

The mechanism of constant rate drying is that of evaporation from a

liquid surface with little interference by the presence of the solid

(Nonbehel S Moss, 1970). The resistance to mass transfer is usually assumed

to be completely in the boundary layer of the drying gas. Though solid may

aTfect the properties of the liquid surface so that the rate of evaporation

is somewhat different from that obtained with a pure liquid, this solid

effect is relatively small. It. often corresponds to a reduction in the

evaporation rate of not exceeding 20% (Nonbehel & Moss, 1970).

The rate of drying in the constant, drying rate period is determined by

the rate of vaporization of the liquid from the drying surface into the main

body of the gas stream. The entire surface of a solid particle tends to

stay at the wet-bulb temperature corresponding to the temperature, humidity

and quantity of the drying gas. If the condition of the drying gas at the

surface of solids remain constant, the surface or wet bulb temperature will

also be constant. Consequently, the partial pressure and humidity at the

11-37

surface will be the saturation partial pressure and the saturation humidity

at the wet-bulb temperature, respectively.

The velocity of the drying gas affects the mass transfer coefficient or

equivalently; the resistance to the mass transfer. It should be noted that

strictly speaking, a true equilibrium at the wet-bulb temperature is not

always assured. However in practical drying situation, it can be assumed

that the actual surface will approximate to the wet-bulb temperature for

constant-rate drying with a sufficient quantity of the drying gas in the

main stream and with heat supplied mainly through convection.

The moisture content at which the drying rate of a product changes from

a constant rate to a falling rate, is called the critical moisture content.

Generally, it increases with the increase in drying rate. It often depends

on the physical properties of the solid, such as shape and size, and also on

the drying conditions. It usually need be measured experimentally.

Approximate values for many industrial solids are available (McCormick,

1973) .

Falling-Rate Drying Period

During the falling-rate drying period the surface of a drying particle

is not covered with a thin layer of water as is the case during the

constant-rate period; the interna] resistance to moisture transport becomes

greater than the external resistance. As the moisture content of a product

falls below the critical point, the drying force decreases along with the

drying rate. Also, a moisture gradient appears within the drying product

and the product temperature rises above the wet bulb temperature.

The movement of moisture inside a drying specimen may occur by various

mechanisms, including liquid diffusion, capillary flow and surface activated

11-38

diffusion, depending on the type of solids. For capillary porous products

like cereal grains, for example, the following mechanisms are suggested

(Brooker et al. , 1981)

(1) liquid movement due to surface forces (capillary flow)

(2) liquid movement due to moisture concentration differences (liquid

activated diffusion)

(3) liquid movement due to diffusion of moisture on the pore surface

(surface diffusion);

(4) vapor movement due to moisture concentration differences (vapor

diffusion)

(5) vapor movement due to temperature differences (thermal diffusion);

(6) water and vapor movement due to total pressure differences

(hydrodynamic flow)

Luikov et al. (1966) have developed an mathematical model for describ-

ing the drying of capillary porous products based on the physical mechanisms

listed above. The model equations are a system of partial differential

equations of the following form:

3x

3^ .9 K„xp

4 V2K12

T V2k]3

P (70)

3T 2 2 2

8l " V K21

Xp

+ V K22

T*V k

23P

< 7 ')

3H 2 2 2

St"- V K

31Xp *

V K32

T'V k

33P

< 72 >

where K,

K22

and K are the phenomena logical coefficients, while the

other K values represent the coupling coefficients. The coupling results

from the combined effects of moisture, temperature, and total pressure

gradients on the moisture, energy and total mass transfer. At the present

time, the phenomenological transfer coefficients are available only for a

very limited number of substances; therefore, Luikov's system of equation

has not yet been widely employed.

In actual situations, many simplifying assumption have been made in

translating Luikov's rigorous model into working equations. For example, in

drying of cereal -grain , the pressure and temperature gradients usually do

not have to be considered. Neglecting temperature and pressure gradients,

the Luikov equation is reduced to the unsteady state diffusion equation

~B~ V

2

«V (73)

where

l'=K n (74)

I) is usually called the diffusion coefficient. With constant value of 11

equation (73) becomes the well-known Fukian diffusion equation

Assuming uniform initial moisture distribution and negligible externa]

resistances, the solution to eqn . (75) for granular particles is (Crank,

1975)

x -x co , 2 2 2P Pe 6 1 n n x ,7~— ' 1 J

1 1 eXP <- -9 » < 7S >

po pe n h"l n

* | TdV) ! (m)1/2< 77 >

p »

It is often feasible (see, e.g., Rrooker et aJL , 19H1) to further simplify

eqn. (76) hy employing only the first term of the infinite series. Thus we

have

11-40

f^ --f

exp«- Kt, ,„,

whore

K —5 '1 (79)

ldPV

The error is less than 5* if the dimensionless quantity Kt > 1.2. The

values of the diffusion coefficient, D, in euns. (76) and (78) are available

for various foods (see, e.g., Chirife, 1983) and for cereal grains (see,

e.g. Hrooker et al. 1981) .

The temperature dependence of the diffusion coefficient is often ex-

pressed in an Arrhenius form, i.e.,

D=Doexp(-E

a/RT)

( 80 )

The values of I) and E arc available for various foods (Mujumdar, 1983),

and for cereal grains (Hrooker, 1981). Since the temperature of the

material changes with time, the diffusion coefficient D in eqns. (76) and

(78) should be replaced by time-average value 5, which, by definition, is

n -\ \ dt (81)

Our discussion so far has implied the assumption of geometrically

similar drying curves for the falling rate period of a given material.

regardless of the initial moisture content of the material and the initial

drying rate. When the diffusion coefficient varies strongly with moisture

content, this approach can not be generalized to different initial moisture

contents or initial drying rates by introducing a dimentionless moisture

content since the diffusion coefficient depends on the absolute moisture

content itself.

11-41

A method for calculating drying rates for the case of moisture-content

dependent diffusion coefficient has been developed (Schoeber, 1978). The

method is applicable to a system in which the diffusivity decreases rapidly

with declining moisture content below the critical value or to a case where

the drying rate is governed by the rate of mass transfer inside the

material. For porous materials, however, the similarity approach has been

shown to be a valid approximation.

For design purposes, usually the data for the average drying rate

rather than the average moisture content is preferred. Differentiation of

eon. (7R) with respect to time and rearrangement of the resultant expression

yields

dx

5T ' K(Xp

Xpe

> < 82 >

dx

Ax l B( 83 )dt p

where

A=-K, B=K-x (841pe ' '

Equation (82) reveals a linear relationship between the average drying rate

and moisture content. This is supported by extensive experimental data and

hence is widely used to approximate the drying-rate data.

Luikov (1968) and Lyaboshits et aj.. (1969) have proposed two other

expressions for evaluating the average drying rate in terms of the excessive

moisture content x - x as in eon. (82). They are

dx , x x ,np ( p pe )

d7~=

n < 85 >

C+l)(x -x )

P pe

and

dx—77 - C'(x -x )+I>'(x -x ) (86)at p pe p pe * '

where C,C',D,])' and n are experimentally determined constants. In drying of

certain materials, like foods (see, Mujumdar, 1983), the drying rate in the

falling rate period is characterized by two distinguishable stages. Results

obtained by various authors (see. e.g.. Jason, 1965; Vaccarezza et al., 1974

and Roman et a!., 1979) indicate the existence of an initial straight -1 ine

portion, hitherto referred to as the "first falling-rate period," followed

by a concave downward or another straight line of different slope, forming

the second falling rate period. The linear relationship between drying rate

and moisture content in eqn . (82) or (83) suggests that the Fickian equation

should be applicable to the first falling rate period. This has been

verified by Vaccarezza et a_l. (1979). They measured internal moisture

distribution during sugar-beet drying and compared the data with the

theoretical moisture distributions predicted by Fick's law. A very good

agreement was observed.

It has been found that in most cases, the moisture diffusivity is not

constant with moisture content in the second falling-rate period (see, e.g..

Mujumdar, 1983). Thus the more genera] diffusion equation, eqn. (73), is

prefered for predicting the internal moisture distribution of this period.

Crank (1956) has outlined methods for determining the functional dependence

of the diffusion coefficient on the moisture content. Several empirical

equations for drying cereal grain and foods in the falling rate period are

also available (see, e.g., Brooker, 1981 and Mujumdar 1983).

11-43

MODELING OF FLUIDIZED BED DRYING

Drying characteristics of a single particle under given fluidization

conditions do not adequately describe the drying process in a fluidized bed

dryer as a whole, since particles are seldom dried individually.

Hydrodynamics of the bed, such as bed expansion, generation and movement of

bubbles and mixing of gas and solids, affect significantly the overall

performance of the dryer. Thus, to describe its performance quantitatively

and mechanistically will require a system of governing equations for the

various processes and phenomena occuring in the dryer. A number of assump-

tions are often made to simplify the solution of such a system of governing

equations; these include

(.1) volume shrinkage is negligible during the drying process;

(2) temperature gradients within individual particles are negligible;

(3) particle-to-particle heat conduction is negligible;

(4) moisture equilibrium isotherm is known.

Modeling of drying operations in fluidized beds are also based on

whether they are batch or continuous. Strictly speaking batch drying is a

semi -batch process where the material to be dried is exposed to a con

tinuously flowing stream of drying gas. In continuous operation, the

substance to be dried and the gas both pass continuously through the bod.

Batch Operation

Batch fluidized beds are used often for small processing capacity.

Capacities of 50 kg/hr or less have been categorized as being "small"

(Viswanathan et al , , 1982). References regarding the operating data are

available (Viswanathan et a}., 1982)

Design procedures for batch fluidized beds have been proposed by

various investigators (see, e.g. Vanecek et aJ . , 1966; Kunni and Levenspiei

,

1969). Most of them, however, are simply based on the total heat and mass

balances around the dryer. The interactions among some of the components

and entities, for example, generation and motion of hubbies and their ef-

fects on transport processes are, generally, not considered. Thus, they are

quite empirical in nature and heavily dependent on experimental data.

Recently, some comprehensive and mechanistic models have been developed

(Hoebink and Rietema, 1980(a) and 1980(b); Viswanathan, 1982; Viswanathan

and Rao. 1982; Viswanathan et al. , 1983; Viswanathan and Rao, 1984) to

describe the drying process in a gas-solid batch fluidized bed. These

models have incorporated the effect of bubbling characteristics and various

mechanisms of transport processes in the bed.

Model proposed by Viswanathan and Rao

As previously stated, Viswanathan and his co-workers have published a

series of papers which eventually have culminated in a comprehensive model

for a batch f luidized-bed dryer (Viswanathan and Ran, 1984). Their model is

essentially based on the three phase theory of f luidization , which includes

'bubble, cloud-wake and emulsion phases (see Fig. 12 for schematic repre-

sentation). The tanks-in- series model is assumed to be applicable to the

emulsion phase; in other words, the emulsion phase is treated as being

composed of a number of compartments, with emulsion gas being completely

mixed in each compartment. Both the downflow and upflow modes (the latter

is illustrated in Fig. 14) have been assumed separately for the overall flow

of the emulsion gas. In the bubble and cloud-wake phases, the drying gas is

considered to be in plug flow. The solids in the dryer are regarded as

11-45

being homogeneous with negligible internal resistance to mass and heat

transfer. They are considered to be throughly mixed in the bed.

Consequently, the temperature of the bed, the moisture content of the par-

ticles and hence the equilibrium surface moisture content of the drying gas

are independent of the position in the bed. Furthermore, the drying gas is

assumed to be at quasi-steady state with respect to the solids in the bed.

This is due to the fact that the residence time of the drying gas is much

shorter than the time needed to dry the solids. Thus, the moisture and

energy transfer in the gas phase in conjunction with the solids is ap-

proximated by that under the steady state conditions. Hased on these

assumptions for the change of moisture and energy in various phases, a set

of governing equations has been derived. The procedure of the derivation is

described below

Bubble phase. A moisture balance around the controlled volume j]

lustrated in Fig. 13 yields (see APPENDIX A)

dxb

Ub Si '

(KbcV Xc V <« 7 >

QlSSJ wakc P.!l22«- Referring to Fig. 13, a similar governing equation

for the cloud wake phase can be derived as below (see APPENDIX B)

dx

(a+/9)f u —p = (K ), (x . x ) (K ) (x x )mf b dz ce b ei c be be b

*- (<M-/S)<1 € )S K (x - x ) (88)f P Pg P c

Emulsion gas. For the case with drying gas flowing upward through the

emulsion phase as depicted in Fig. 13. a moisture balance around compartment

i results in (see APPENDIX C)

11-46

[1-y i+°^ )] (xei

xin(i)>

H u e . —, ^r—b

e mf (zi

" zi-i>

ri *b

< ]+cr+/e) j ," r (l-« f )S K (x - x .) t (K ) <x - x .) (89)*

bmf p pg p ej ce b c ei

x in eqn. (89) is the average moisture content of the gas in cloud

in contact with the ith compartment- It is defined as

J' *_*« (90)c z. - z . , J c

l l-l zj 1

Equations (87), (88) and (89) can be rewritten in dimensionLess form as

dXb

—77 - a (x x ) (9]

)

dZ c b v'

dxc

~E= a

l

(Xei *c>

4 a2(x

b- V

+ a3(X

p - V < 92 >

(xei"

xin(i) »

W~7T^r " 34(X

P - Xei>

h a5(x

cXei» <

93»

where

be b f ce b f

ub

al ' (o^)e

Bfub

2 (a^)6mf

ub

3*

<^>VUb

(94)

(95)

[l-«b(l+a+/8)]— (l-e )S K (B(4 «b

mf p pg h f

ce b h f

5 " [1-6. (l +cr+/8)]u f ,(96)

D e nit

z Êliminating x from eqn . (92) by resorting to eqn . (91) yields

2d x, d x.

b . b

dz dZ

where

b

a +a +a„â. o!23

1 a„a.

3

The solution to eqn. (98) is

xb

=

b~(xet

+bV ' *iexpt

*l(E"2

l-i)1+d

aIMipt>

a(Z"Z

i-i )J '

z.<z<z.

where

and

-b, * (bj^b,) - 5

(97)

b x + b *• b x." x . + bx 08)2 ^.,2 1 j, o b ei p\->v>

(9-1)

(100)

b2

"^ (1"D

(102)

!_!' 1<1<N (103)

(104)

Zo

= " (105)

dl

and d2

in eqn - <-103 > are Constants to be determined from the boundary

conditions for each compartment. Combining eqns. (91) and (103) we have

11-48

1 * '"lxc

=b~

(xei

+bxP »

+<1+ 5-' d

iexe[<V z-Vi>]

2< 1+ — )<i

2exp[(m

2(Z-Z.

_1)], Z <Z<Z., Ki<N (106)

Substitution of eqn . (106) for x in eqn . (93) results in

Xer Xin<>) "{!£•«< V'VWei

,a4fa

5 ^" zi *i-i>«;

V dl

' as(lf

ir)-i;fexP<B izi) «p<v, _

t)J

m, d

* a5(U

S_)

ir r ';xp(m2Z

i) " exP<"

8zj_ 1

)] <j07 )

Rearrangement of this equation yields

Xei

= (Xin(i)

f fl

dl

+ f2d2

+ B"p)/(1 +S) (108)

where

a5

mi

fl

= 5" (1+ r-Hexp(m Z)

- exp(m Z )J (109)2

x i i

a m,

f2

=

ST<lH ^-; Hexp(m

2Z.) expfn^Zj^}] (110)

e - (a4,a

5t.)

( z. :w ] (m)

From eqns. (103). (106) and (108), we obtain, respectively

Xc

= Xb

+ V"l(iieXPtVZ~Z

i-t>]+ m

2d2K*plm

2i7-' Z

i-l )]) (U2)

11-49

in(i) ,, .1 .*

+ di

<exp[m.

(z - z.i-i

)] +

iTTTTiT 1

- d 2t exp[.2(Z-7.. _,)! ^T^,) (113)

The continuity condition, stating that the mass flow of moisture in the

emulsion gas at the entrance of each compartment is equal to that at the

exit of the preceeding compartment, comprises the boundary condition for

each compartment, except for the first one (see Fig. !4). In other words,

"em^yî'înU, " "ed-l^-Vî-lêt-l

for i > 1 (114)

The boundary contion for the first compartment is

at Z=0 (115)

The compartment size Ah. and other parameters in eqns . (108), (112) and

(113) art; determined from the relationships listed in table 3. The moisture

content of the outlet gas is given by

Vout ' WXb! 2-1 ' (°^ )e

mfXclz.1 ]

+ [1"*b

(I -a^ ),Ueemf

XeN < 1161

Solid pha se. A moisture balance around the entire solid phase results

in (see APPENDIX D)

% p%\ ,

"dT= ~W~ (x

in" x

out> I 1 "'P

The appropriate initial condition is

xp

= xpo

at t - (118)

11-50

The corresponding energy balance is (see APPENDIX E)

dT "Vtr ,

dT WT [(Xin

Xo„t>*0 '

(Tin

T)Cg

! < 119 »

P Pu

The initial condition for this equation is

T * Tq

at t = (120)

Equations (117) and (119) can also be rewritten respectively as

% pVtit " ir~ E(x

i„- y ( 121 )

p

dT "Vtdt

=M c 'în^p'ô T

>*tn"" uB

P P

,XIn-

Xn»

ETn+

<Ti„-T)cJ (122)

where

(x. - x J„ in outE = —

(123)(x. - x )

in p

It has been assumed that the following relationship exists between the

equilibrium surface moisture content of the gas and the moisture content of

the particles

Xp

'A X

p("«)

where A is the equilibrium constant to be determined. To incorporate the

effect of temperature on the equilibrium constant, it is further assumed

that

A = BT (125)

where B and m are constants. Now eqns . (121) and (122) become, respec-

tively,

dx PV„K3jE.-fi B„ btV, (126)

p'

II 51

at - srr [(xin

- BT XP»% H

< Tu,

- T>V (127)P P E

Equations (126) and (127) are coupled differential equations, which can only

be solved numerically. If the variation in T is negligible compared to

variation in x eqns. (126) and (127) can be solved independently.

*In the constant rate drying period, x is the saturated moisture con-

*tent of gas at the surface of the particle, x . which is constant. It

po

follows from eqn . (121), that

P"o

At

Xn

=\, ,

" ~S E (x x. )t (128)p po M po in

P

Equations (108). (112), (113), (116), (126) and (127) with the ap-

propriate initial and boundary conditions constitute the governing equations

of the model. These governing equations need be solved simultaneously.

Starting with t-0, T=T . x x and x -x , the use of eqns. (108), (112)o p po p po ' x '

and (113) for all the compartments, beginning at the one at the bottom gives

the initial variation of moisture content of the gas in different phases

along with the bed height. The moisture content of the exit gas x isout

evaluated from eqn. (116). Equation (126) and (127) can then be integrated

to yield the temperature and moisture content of the particles within the

time interval dt. These values are used for calculating the temperature and

moisture content of the gas in the same time interval dt. Repeating the

same procedure, the time-dependent profiles of temperature and moisture

content of the particles and the gas can be obtained.

11-52

In the case where gas to particle mass transfer resistance is negli-

*gible, that is x =x

g=x

c. a simplified two-phase (emulsion-bubble phase)

model can be employed. The governing equation for the bubble phase eqn.

(91) becomes

dxb

dZ op b

The solution to this equation is

a (x*-x) (129)

xb(z)=x

p. |x

b(z.

l)-x

pj Cxp(-e.(z-z._ ]) ]

i > 1 (130)

wtiere

e-

=( KK )uH «'' u u' evaluated in ith compartment

l be b t b

From eqn. (130), the moisture content of the gas at the outlet of the ith

and the i-lth compartments are. respectively, given by

W - V IV*i-i , "*p ,*"pt"*i <Vii-i )I (131)

V2i-i>=v Cx

b(zi-2 )"V e*pt -e(zt-rz i-2 )] (132)

Combination of eqns . (131) and (132) yields

VV"V <V*t-* )"V ]e*pt --e! i

,z i-r zi 2

)ei

(zi"

zi t» (133 >

Following this recurssive relationship, the moisture content of the outlet

gas in the bubble phase from the top compartment (i-N) can be evaluated

from

11-53

V zN)=x

b(1,=V (xiirV exp! ^. sW (134)

Then the moisture content of the outlet gas at the exit of the bed is ob-

tained by combining gas flow from the bubble phase and emulsion phase

lVout=Vb [x

b(1,Ma^» e

mfXelt|1A (Ka^ )1Vmf X

e(1351

where the parameters should be evaluated at the Nth compartment. From

*x -"

e=x

ieqns. (92) and (93) can be combined into one governing equation

for cloud wake phase and emulsion gas. The resultant expression takes the

form

x.=x=x, 1 < i < N (136)ei c e i^«i

and for the first compartment

(x_-x )

Ue[1-V 1+a^ )lemf-%f- ' ( V»b (x

b- x

e» < 137)

-1 i faxb

= T Jxbdz

< 13a >

The governing equations for the solid phase eqns. (126) and (127)

remain invariant since no new assumptions are introduced in the derivation.

In summary, the governing equations for the simplified model consist of

following five equations

Vêi^VW êi 1 '^' 1 Vî-l" (139 »

l!

oXout'Vb fx

b(,) * (a^ )e

mfXeN

l

+ " -V 1+0,""V-f xeN

(14("

x -xe o

•,tl-*k(l^)|ll(T-. (KhJ h<<-XJ ("D

11-54

dx PU A

d^= IT- '"inôut' < 142 >

P

dT ire1 [(X

in-Xo U t

)V <Ti,r

T)cE

J ll43 >

p p

The calculation procedure Is similar to that described for the general

model

.

The criterion adopted to determine the validity of the simplified model

is (Viswanthan et al. . 1983)

Nr

* 50 (144)

where

H

N = S K (1-6 „)— (145)r p pg mf U U«oi

o

The restrictions on the application of the model are the assumptions

that the temperature and moisture content inside the particles, and the

temperature of the bed are uniform. The former may be true only for small

particles. Also, the gas-solid transfer mechanism in the emulsion phase may

differ from that in wake region. The solids in the wake region are con-

tinuously washed out from the wake and replaced by the fresh emulsion

solids. Thus, the moisture transfer between the solids and gas in the wake

is determined by the exchange rate between solids in the wake and those in

the emulsion phase; in other words, it is determined by the contact time

between the solids and gas in the wake. In the proposed model, however,

solids are treated simply as an entjtjy and the mass transfer between the

gas and solids in the emulsion phase and that In the wake region are con-

sidered to be the same.

11-55

Model proposed by Hoebink and Rietema

The gas-solids transfer mechanism in the emulsion phase and that in the

cloud-wake have been investigated separately in the model proposed by

Hoehink and Rietema (1980(a). 1980(b)). The former is considered to be

affected by the profile of moisture content inside a particle, that is. the

transfer mechanism is diffusion controlled. The latter is assumed to be

governed only by the flow through a thin concentration boundary layer inside

the particle because of the relatively short residence time of the gas in

the cloud.

The model is based on the three phase theory of f luldi zation . For

simplicity, a uniform bubble size is assumed throughout the bed. The con-

ventional bubble-cloud model is modified through several additional

assumptions: 1) the flow of both gas and solids through the cloud is con-

stant and equal to the flow at the bubble equator; 2) the gas and solids

pass the cloud in plug flow; 3) the zone of the cloud where exchange of

moisture takes place between cloud and solids, and between cloud and emul-

sion gas is restricted to the hatched area, confined with n/A < 6 37T/4 (see

Figs. 15 and 16). The solids are assumed to be perfectly mixed. The emul-

sion gas is assumed to reach thermal and concentration equilibrium with the

solids within a negligiblily short distance from the distributor. The

solids are considered to be dried only in the falling rate drying period,

during which moisture transfer is mainly controlled by diffusion process

inside the particles. The Fickian diffusion mode) is assumed adequate in

describing this diffusion process. The procedure in deriving the model is

described below.

Cloud -wake phas e. There are three different regions in the cloud wake

phase. The first region is confined by < 9 «; n74 , through which a through

11-56

flow of gas from the bubble phase passes. Since the moisture exchange does

not occur in this region, the moisture content of the cloud in it equals

that of the gas bubbles. The second region is represented by hatched area

in Fig. IS, where moisture exchange takes place between cloud and emulsion

gas, and between solids and gas In the cloud. Due to the moisture exchange,

the moisture content of the cloud in this region increases with e. The

third region is bordered with 3w/4 < 6 < n , where through-flow gas reenters

the bubble at its base. Since there's no moisture transfer as in the first

region, the moisture content of the cloud in this region remains constant.

For the second region, a moisture balance around the controlled volume

depicted in Fig. 17 yields (see APPENDIX F)

Q 3xKC ("2 2 3 3 *

ilni W " an,cKe

( "e-*e , *3r< "c"Rb>(Wf'WW (146 »

The appropriate boundary condition is

V xb

at 9J (147)

Assuming the mass transfer coefficient between cloud and solids K to becp

constant, integration of eqn. (146), subject to eqn . (147), gives

exp { ~-[k\ + ;(8V)(l-« ,)K S ](cose - &)\ (148)I Q c c 3 c b if ps p' 2 J

_ „ ex

K,x +X„x -(X +X )x,1 e 2 p 1 2 b

where

2irR2K

\ ' —Q^< 149 >

fir<R? -R^)(l-e .)K s, 3 c b if cp pX2

= q ~(150)

11-57

Since the moisture content of the emulsion gas is in equilibrium with the

*surface moisture content of the particles, it follows that x =x Thus

e p

eqn. (148) can be rewritten as

*

p b

The moisture content of the gas in the through flow reentering a bubble at

its base is denoted by, x . It is defined as

3x =x at e = -n ( 15? 1en c 4 ' 13i l

Using the relationship between x and x and eqn. (52), we obtainc en

*

en p r 2/277-, 2 13 3 1—

f

-P{ - -HW S'VV'^mf'W 1

(153)

b pgc

An energy balance around the controlled volume indicated in Fig. 17 leads to

(see APPENDIX G)

q pc. 3Tgc g g c

s i me ae

The boundary condition is

( 1 55

)

Integration of eqn. (.154), subject to eqn. (155), yields

Ve^2 Tp

-(W T

c

Ve^2ywrb

R3-R

3

eXp[Q-f

[

c-(Hce

Rc+^ (1 ^mf)V P

)<C ° S9- f )J (156)

where

11-58

m) | ir<»*-R*)U-6 ,)h s

P\ T. '

P2 T

Ope.in

It is assumed that T -T , eqn. (156) then reduces to

T -T R -R

v%= expiv^ [HceKc2+ "V (1^

m f)h

PsP1(cose " f]) ,158)

The temperature of the drying gas reentering the bubble at its base, T ien

related to T throughc

Tm - Tc

at 6 = 3/4 IT (159)

Thus from eqn. (158), we obtain

T -T _ R3

-R3

en p . -2/2w , 2 e h

v\T= expt

vy^ '"--"~ u -e-f

)hP9P>

] (160)

Bubble phase . A moisture balance around the controlled volume shown in

Fig. 18 yields (see APPENDIX H)

oXn

6 u a w a— = n Q (x -x, ) (161)b b dz gc en b \ xv *i

where n is number of bubbles per unit bed volume. The boundary condition

for eqn. ( 161 ) is

Xb

= xin

at z=0< 1R 2)

Substituting for xgn

in eqn. (161) from eqn. (153) and integrating the

resultant equation gives

x x nQ zE; - exp[- —1£- <!-£ >J (163)

x . -x b bin p

where

* "exp( "IT 1RX* i « R

c-Rbl(H^Vp 1! (IM)

EC

In similar fashion, an energy balance yields (see APPENDIX I)

Vb dT ' nV Ten-V < 165 >

with the boundary condition

Tb

- To

at z = (166)

Equation (165), combined with eon. (160), results in

T -T nQ z—£--«Spt-_S£. (i-^j] (167)in p b b

where

3 3R -RV^Vo^-tVp 11 ,168)

gc

The temperature of the exit gas T can be determined from an eneresout

balance over ail the outlet streams. Neglecting the temperature difference

between cloud and emulsion phases, it can be approximated as

Tout

' <l-)Tp,sT

b(M

f) (169 )

In the expression, s Is the fraction of gas flowing through Uie bed as

bubbles, and is evaluated through

b

U "bs - — <S, (170)

11-60

Since the bed temperature is assumed to be uniform, the moisture content of

the exit gas. x can be related to T via an energy balance over the

entire dryer. It states

T (x -x. ) « (c +x. c )T. - (c 'x c IT (171)o out in g in w in g out w' out !*«»<

EEl!l2i™_gS§ From the assumptions at the outset of this section:

and

T =Te P

the state of the emulsion gas is determined by that of the particles.

Solid particles . Two mechanisms for the mass transfer between gas and

solids arc considered to exist. In the wake region, solid particles are

exposed, for a very short period, to a gas concentration which is lower than

that in the emulsion phase. The concentration profile inside the particle

wouldn't be able to respond to such a sudden change in the gas

concentration. Thus, it is reasonable to assume that the mass transfer is

controlled by the resistance in the gas film surrounding the particle and

the concentration difference across it. On the other hand, because of

the relatively long contact time between particles and the emulsion gas. the

mass transfer is likely to be affected also by the diffusion of moisture

inside the particle. The combination of these two mechanisms gives an

overall description of mass transfer between the solids and the drying gas.

a). Solids in the cloud-wake phase. The transport of moisture inside the

particle follows Pick's law of diffusion. For constant diffusivity of

moisture and a spherical particle, it nan be rewritten as (see, e.g.. Bird

et aK , 1960)

% 3,2 %,

1 1-61

The appropriate initial conditions are

t=0 x =x < r < Rp po p

and the appropriate boundary conditions are

ax

t>o3r

3x ax

-I) —E 4^RZ=V -£

3r p p 3t

r=0

r=R

(1T3)

(174)

The second boundary condition states that the flow of moisture out of a

particle through its surface is equal to the increase in the moisture con-

tent of the gas surrounding it. The solution of eqn. (172), subject to

eqns. (173) and (174) is available (Crank, 1956; Carslaw and Jaeger. 1959).

It takes the form

x 2ER x

K . _£2 . P p o

p E + l 3r

2 4 2m E q.+3(2E+3)q.+9Z exp(-q F )-~ —

i = l E q.+9(E+l)qf

where

sin(q.r/R )sin q.

Dt/R , EP 4 „3

-7TR m3 p

and q.'s are positive roots of the equation

(175)

(176)

(3+Eq )tan q. - 3q

.

(177)

m in eqn. (176) is equilibrium constant relating the moisture content of the

drying gas and that of the particle. The moisture content of the particle

in ultimate equilibrium with the drying gas, x can be evaluated throughpe

eqn (175) by letting t -». It is

pox » —

—

pe E+l (178)

The coefficient of mass transfer inside the particles. K . is defined asPi

3x

p

Combining eqns, (175) and (179). we obtain

2

" 2q

j

s ^p'-i/oi'T!kp.

rp . ..

] E 'V 9 '^ 1 '

Shpl° ~D~ "

3 >' m < 180 '

v /2 „ i 1

51 exp(-q,F )

i 01 2 21 B qf+9(E+l)

So far, the resistance in the gas film has been neglected. If its effect is

incorporated into the overall mass transfer coefficient. K we obtaincp

1 _ 1 1

k~ = r +mlT

-< 181 >

PE g pl

Kpl

iS determined by eqn. (180) with t=t , the average residence time of

solids in the cloud with volume V We consider the solids to pass through

the cloud at the same velocity as that for the through-flow gas. t is,

therefore, determined by

t

Ve z

n(Rc-

RiXf J£„,

1" Q"

=3* ~

Q

lT ':0se) "» a )

go E

and V is enclosed between angles w/4 and 6. In case where

1/m. (183)

which implies 6 0.5, «K can be approximated by (Hoebink and Rietema,

1980(b)

)

R

mKpl

=3t~ (184)

Combining eqns. (181) and (182), we obtain

1 1 -63

K R

K - g P

pg R +3K 1:,

P g 1

Substituting for t in eqn. (185) from cqn . (182) yields

(R3-R?) K _

K = K [ 1 +27T—^ (1-e )-M (<1 - coa e)]

pg g Q mf R 2 '

K P

(185)

(186)

Recall that in deriving eqn. (151), we assume k ^ to be constant. It could

now be replaced by its 9-average value, which can be obtained through in-

tegration of eqn. (186). A better way of doing it, however, is to go back

to the original differential eqn. (146). Substituting for k in eqn. (146)pg

from eqn. (186) and following the same procedure as that in deriving eqn.

(151), we obtain (see APPENDIX J)

V XPR

/m

2rrRc

\(f-- COS9)

(K3-R^) K _

U mj R 2gC p J

( 187)

X can be evaluated from the above equation by setting 6 = -w . Equatic

(163) can now he rewritten as

Xb

~ XPR/ir.

in PR/m

nQ (1 -fi )zg m

b b

(188)

in which

2/2>rR

fl = exp(- -K 1 + 2j2rr

3 3(R -RT

c bK

mf R

-1

(189)gc L

vgc

Taking an average of the moisture content of the gas from the emulsion phase

and bubble phase at the exit, based on their flow rates, we obtain the

moisture content of the outlet gas as

II -64

Xout - I 1

'8' "F * *VV

Combination of eqns. (188) and (190) yields

out PR

in PR

s expQ (1-/8 )H,gc m f

b b

(190)

(191)

b). Solids in the emulsion phase. The assumption of the complete mixing of

solids and gas in the emulsion phase implies that the drying characteristics

of particles can be represented by that of one single particle. Since all

particles are assumed to be identical, the gas flow to which an individual

particle is exposed equals the total gas volumetric flow divided by the

total number of particles in the bed. The average moisture content of the

particle tends to reach the ultimate equilibrium with the emulsion gas

during its relatively long contact time with the emulsion gas. The profile

of moisture concentration inside the particle can be obtained by solving

the following Fickian diffusion equation;

3t 2 3rl

3r' (192)

with initial condition

x =xp po

for t-0 < r < R

and the boundary conditions

3x

and

3r

3xP

3r

for t > r=0,

for t >

(193)

(194)

(395)

where A is the average moisture flux from the particles and is given by

47TR<f>

- Q (x -x. )p p p out in (196)

11-65

Substituting for x from eqn . (190) yields

(l-Y)Q xE (_™

m in (197)

in which

Y = s expr q (i-/8 )n.

igc m f

Vb (198)

Q in eqn. (197) is the gas flow to which an individual particle is exposed.

It js determined as

Q A,

'p N

t 1 mf b 3 p

4irR3U

P o

3Hf(W

mf»(1-V

(199)

If we define the specific surface area of the particles based on unit vol

of the bed , S , , asph

sP b

= (wBf

)(i-VdVP s

Equation (197) can now be rewritten as

(l-Y)U

*P "

"V^- IV - *in>

(200)

(201)

K (x /m-x. )

g PR in

Kg

.which is defined in eqn. (201), is the average mass transfer

coefficient. Substituting for <*>

pin eqn. (195) from eqn. (201), the boundary

condition, eqn. (195), can be rewritten as

3x x-0 -E . j< (J* v ,or g m in

t > 0, r=R (202)

11-66

Equation (195), subject to eqns . (193) and (202) can be solved by means of

Laplace transformation (I.uikow, 1968), which takes the form

x (r) -mx.p in

x - nx.po in 1

sin(u.r)/H

)i. r/Rexp(-fJ.Fo

2) ( 203

)

with

Fo„ = Dt/K

'

2 p

u in eqn . (203) are the non zero positive roots of

( J-B)tanji.=f*.

where

(204)

(205)

K R

B "n,D (206)

which is the overall Hiot number. The mass transfer coefficient for the

particle in the emulsion phase, K is defined as

3x

p

(207)

where x is the average moisture content of the particle. From eqn. (203),

we obtain

K ,,R I ^TiB" -B

p2 C

rj: exp (

"iFo

2'

3 n 2(208)

~2~y<— BXP ( î F°2'1 f.l' + B -B

i

From eqns. (202) and (207), the rate of drying of solid particles in the bed

nan be expressed by

11-67

dx „ 3x

dtl

d * " 3r ir = R'

P s p

, 6 w pR

a d> m inp^s

(d~T)(x

Pxpr» < 209 )

p^s

Eliminating x from the above equation, we obtaint K

dx"

df " ldV ,Ro

(,1n-Km l

< 210 >o p inP a

where K is the average overall mass transfer coefficient, defined by

1 1 m— " — + — (211)K p2 Ko g

If we define the drying efficiency as

(212)

x -mx

.

p 3 11

equation (210) can be rewritten as

" df " (dV'V (213)P 8

The initial condition is

X=l for t=0 (214)

Integration of eqn . (213), subject to eqn. (214), yields

X . exp[- (^t-)J

Kodt]

< 2]5)p S O

11-68

The governing equations of the model are eqns . (167), (169), (171),

(187), (188), (191). (203), and (215). To obtain the profiles of tempera-

ture and moisture contents of the gas and solids, first x is determined

from eqn. (203). Then x (z) can be evaluated through eqn . (188). With

xb(z), and x

pRknown, x (6) can be calculated from eqn. (187).

Subsequently. the outlet gas concentration is obtained with the aid of eqn.

(191). It follows that the outlet gas temperature can be determined by eqn.

(171). Solving simultaneously eqns. (154), (167), and (169) results in Tb

T,.'

and Tn

'thereby yielding the overall performance of the hatch ftuidized

bed dryer.

II -69

Continuous Operation

In contrast to a batch fluidized-bed dryer whose content of solid

particles is fixed, the solids in a continuous fluidized-bed dryer are

constantly added to and removed from the bed. Though a single particle can

be viewed as a batch dryer, the performance of a continuous fluidized-bed

dryer confining a multitude of particles can not be evaluated by merely

considering the behavior of an individual particle. This is due to the fact

that the state of the drying gas by which drying characteristics of solids

are affected depends also on the flow of solids. Consequently, the modeling

of a continuous fluidized-bed dryer is even more complicated than that of a

batch fluidized-bed dryer.

Relatively little has been published on the modeling of continuous

fluidized-bed drying. In the few existing models (see, e.g., Vanecek et

al.,

1966; Kunni and Levenspiel, 1969; Palancz and Part.. 1973). it is often

assumed that the bed temperature: Is constant and Lho outlet streams are in

thermal or concentration equilibrium. The fluid mechanical behavior of the

drying gas is considered homogeneous; In other words, the drying gas is not

partitioned into different phases of the fluidized-bed, such as the emulsion

and bubble phases. Instead, these models are based on the overall heat, and

mass balances over the dryer. They do not take into account the intricate

transport processes of moisture and energy among various phases, and thus

these models are often restricted to specific applications.

Mode] proposed by Palancz

Recently, a mechanistic mode] has been developed for continuous

fluidized bed drying (Palancz, 1983). Inessenced, the proposed model is

based on the two phase theory of f luidization . The bubbling behavior and

11-70

its effect on the performance of the dryer are characterized by the

visualization that a soiid-free bubble phase exists in the bed and that this

phase constantly interacts with the emulsion phase. The moisture in the

bubble phase is enriched by the emulsion gas circulating through it. As a

result, the moisture content of the gas bubbles increases along the bed

height. The energy transfer accompanied by the moisture influx to the

bubble phase tends to increase its enthalpy. Nevertheless, the temperature

difference between the bubbles and emulsion gas generates a net heat flux

from the former to the latter. This temperature difference is due to a

decrease in the temperature of the emulsion gas caused by supplying heat to

the solids for vaporization of moisture. Consequently, the temperature of

the bubbles also decreases during their passage through the bed.

The bubbles are considered to be of the same size and their movement to

be in plug flow. This implies that the bubble breakage and coalescence are

neglected. The emulsion gas and solids are assumed to be completely mixed.

The solids are also assumed to he homogeneous with negligible internal

resistance to the heat and mass transfer. The moisture evaporated is as-

sumed to pass through a thin stagnet gas film surrounding the solids before

it reaches the main stream of the emulsion gas. The resistance in this gas

film and the concentration difference across it govern the moisture migra-

tion from the solids to the emulsion gas.

By superimposing the mechanisms of moisture and energy transport in

various phases presented in the proceeding paragraphs, the following govern-

ing equations have been derived (Palancz, 1983)

(K ).SXb

= Xe ' <Vx

tn,expt

GZl

< 216 >

11-71

P U eg >f , .

Z T ( x "x- )H. 6. e in

t b

VWb <W +<;r> \ VW

P dx

1 4 — X

dt l

d ' pgl

ps p

P„ PO

T. = T + (T -T ) exp[ —— 6 I (21H)b c o e 'L

p u, c b (<=!»)

s* b wg

C p U „—*—a( T -T 1

b f

(1-6 )(1-<5)'WW 4

'T2

' 4 ! h <T ~T

>-"<

<x -* >C T Ibe o e b a f P e p pg

(

p e »g e

- hw (VT

e» ST (219)b

dT

p (c +c x )jjt

s p w p dt

l + ^x6

d Z [hn (V T

,J K.,„K x XT *C T -c T )j (220)QD P Pep pg p e o Wg e w p

v'

P,., P°

P

The solution of the above system of equations determines the temperature and

moisture content of solids and gas in the different phases. These values

can be used to evaluate the outlet gas temperature and moisture content as

well as the average temperature and moisture content of the solids leaving

the dryer.

The Palantz's model assumes a constant specific heat of the drying gas

throughout the drying process. This implies that the change of moisture

content in the gas, or to he more precise, the change in the moisture con-

tent of the bubhles is negligible; this contradicts the plug flow postulate

for the bubble phase or, in other words, the governing equation of the

mode], eqn. (216).

11-73

NOTATIONo

A. cross-sectional area of the bubble phase, m

A cross-sectional area of tlie cloud-wake phase, m

A cross -sectional area of the emulsion phase, m

Archimedies number, dimensionless

specific heat transfer surface of the dryer wall, ml

c specific heat of dry gas, kJ kg °C

c specific heat of particles, kJ kg °CP

specific heat of water (liquid state) kJ kg »C

specific heat of wet gas, kJ kg °C

2 - '

diffusion coefficient, m s'

!',. diameter of bed column, m

d. bubble diameter,

d particle diameter, m

E activation energy, KJ kmol'

Hf

fluidlzed-bed height,

H - bed height at minimum f luidization

,

HT

overall height of the tapered section in a tapered fluizied bed. m

KPit

11-74

H. volumetric heat transfer coefficient between the bubble and cloud-

»be

volumetric heat, transfer coefficient between the bubble and emul-

sion phases

H volumetric heat transfer coefficient between the cioud-wake region

and the emulsion phase, Js m °C

H.j, overall height of the tapered bed section, m

j, heat transfer factor, dimensionless

j mass transfer factor, dimensionlesso

K))c

coefficient, of gas interchange between the bubble and cloud-wake

1

regions , s

K1]e

coefficient of gas interchange between the bubble and emulsion

. -1phases . s

Kce

coefficient of gas interchange between the cloud-wake region and

the emulsion phase, s

Kc

mass transfer coefficient between the cloud-wake region and the

emulsion phase based on the area of the interface nT2s

"'

K^ mass transfer coefficient in the gas film

k thermal conductivity of the gas. Jm °C

gas-particle mass transfer coefficient, m s1

1 length of tapered bed. m

lQ

length of bottom of the tapered fluidized bed, m

!• top length of the tapered bed, m

M molecular weight, kg kmol

AF pressure differential, N m

11-75

APmf

pressure differentia) at minimum f luidization condition, Nm

Q„_ volumetric flow rate of through-flow gas per bubble, ra"s&c

Nu Nusselt number, dimensionless

Nu particle Nusselt number, dimensionless

R radius of the particle, mP

R Reynolds number, dimensionless

Re Reynold's number at minimum fluidizatJon , dimens i on] essmr

R© Particle Reynold's number, dimensionless

r. inside radius of the fluidized bed, m

r outside radius of the bed, mo

Sc Schmidt number, dimensionless

Sh Sherwood number, dimensionless

St Stanton number, dimensionless

S specific surface of solids, m

t t ime , s

T temperature, *C or °K

Tb

temperature of bubble c phase. °C or °K

T_ temperature of bed. °C or "K

Tc temperature of gas cloud. °C or °K

Te temperature of the emulsion gas. "C or °K

T.n

temperature of gas at the inlet. "C or °K

T particle temperature, °C or °KP

Tw

wall temperature, °C or °K

11-76

uoc

Db

critical fluidizing velocity in terms of the superficial velocity

at the bottom of the tapered fluidized bed, is

linear velocity of bubbles, ms

linear velocity of emulsion gas, ms

superficial gas velocity in the bubble phase, based on total

cross- sectional area of the bed, ms"

un] r

superficial gas velocity at minimum fluidilzing conditions, msJ

°t

terminal velocity of a falling particle, is"1

U particle velocity, ms

W width of the tapered bed

x spacee coordinate, m

XD

moisture content of gas bubbles (dry basis), dimensionless

xc

moisture content of gas cloud (dry basis), dimensionless

xe

moisture content of emulsion gas (dry basis), dimensionless

x^ moisture content of gas at the inlet (dry basis), dimensionless

Xout

moisture content of gas at the outlet (dry basis), dimensionless

xp

moisture content of a particle (dry basis), dimensionless

«xp

moisture content of the drying gas on the surface of a particle

(dry basis), dimensionless

Xpe

equilibrium moisture content of particle (dry basis),

dimensionless

y mole fraction of non-diffusive component, dimensionless

11-77

z space coordinate, m

GREEK LETTERS

If heat of vaporization. KJ keo

s^

fraction of the fluidissed bed consisting of bubbles, dimensioniess

€ void fraction of the bed, dimensioniess

£e

void fraction in the emulsion phase, dimensioniess

6mf

Void fract]on at minimum fluidizing conditions, dimensioniess

^ viscosity, kg m s

2 -

1

v kinematic viscosity, m r

-3P density of gas, kg m£

-3of

density of fluid, kg m

_3Pp density of particle, kg m

cs Stefan -Bo] tzmann constant

(J angular velocity, s

$ effective emissi vi ty . d i mens ion Jess

11-78

LITERATURE CITED

Adams, R. I... Coupled Gas Convection and Unsteady Conduction Effects inFluid Bed Heat Transfer Based on a Single Particle Model, Int. J. HeatMass Transfer, 25, 1819-1828 (1982).

Arastoopour, H. and D. Gidaspow, Vertical Countercurrcnt Solids Gas Flow,Chem. Eng. Sci.. 34, 1063-1066 (1979).

Bakker, P. J. and P. M. Hcertjes, Porosity Distributions in a Fluidized Bed,Chen. Eng. Sci., 12, 260-271 (1960).

Balakrishnan, A. R. and I). 0. T . Pe i , Fluid-Particle Heat Transfer in GasFluidized Beds, Can. J. of Chem. Eng., 53 231-233 (1975).

Barker, J. J., Heat Transfer in Fluidized Beds, tnd. Eng. Chem., 57, 33-39(1985) .

Baskakov, A. P., B. V. Berg, 0. K. Vitt, N. F. Fi 1 ippoviski , V. A.

Kirakosyan. J. M. Goldobin and V. K. Maskaev, Powder Technol., 8, 273-282 (1973)

.

Beek, W. J., Mass Transfer in Fluidized Beds, in Fluidization ed. by J. F.

Davison and D. Harrison, pp 431-470, Academic Press, London, (1971).

Bhattacharyya, 0. and 1). C. T. Pei , Fluid-to-Partide Heat Transfer inFluidized Beds, Ind. Eng. Chem. Fundam., 13, 199-203 (1974).

Bock, H. J., Heat Transfer in Fluidized Beds, Proceedings of the FourthInternational Conference on Fluidization, Kashikoiima , Japan (May 29June 3, June 3, 1983)

.

Bokur. n. H., C. V. Wittmann and N. R. Amundson, Analysis of a Model for a

Nonisothermal Continuous Fluidized Bed Catalytic Reactor, Chem. Eng.Sci . , 29, 1173 1192 (1974) .

Borodulya, V. A.. V. L. Ganzha , S. N. Upadhyay and S. C. Saxcna, HeatTransfer from In Line and Staggered Horizontal Smooth Tube BundlesImmersed in a Fluidized Red of Large Particles, Int. J. Heat MassTransfer 23, 1602 1604 (1980).

Batten']], J. S. M., Fluid-Bed Heat Transfer, Academic Press. N.Y. (1975).

Botterill, J. S. M., Y. Teoman and K. R. Vuregir, Temperature Effects on theHeat Transfer Behavior of Gas Fluidized Beds, A1CHE Sym. Ser. 77 (208),330-339 (1981).

Bradshaw, R. I), and V. E. Myers, Heat and Mass Transfer in Fixed andFluidized Beds of large Particles, AICHE, 9 (5), 590-595 (1963).

Brauer, H., J. Miihle and M. Schmidt, Untersuchungen an einer mehrstuf igenRieselboden-Wirbelschicht, Chem. Ing. Technik, 42, 494-502 (1970).

Brit. Chem. Eng.. Drier for Sticky, Granular Materials, p. 514 (1961).

11-79

Brooker, D. B., F. W. Rakker-Arkema and C. W. HaJ], Drying Cereal Grains,The Avi Publishing Company, Westport, Connecticut, (1981).

Carslaw, B. S. and J. c. Jaeger, Conduction of Beat in Solids, p. 240,Oxford University Press, London, (1959).

Catipovic, N. M., G. N. Jovanovic and T. J. Fitzgerald, Regimes ofFluidization for Large Particles, AIChE J. 24, 543-547 (1978).

Chang, T. M. and C. Y. Wen, Fluid-To-Part icle Heat Transfer in Air-FluidizedBeds, Chem. Eng. Prog. Symp. Series, 62 (67), 111-117 (1966).

Cherrett, N., A Fluidized Bed Dryer/Cooler for Polymer Chip, Brit ChemEng. (Dec. 1964).

Chirife, J., Advanies in Drying, vol. 2, ed . by A. S. Mujumdar, Chapter 3,McGraw-Hill, Washington, (1983).

Cranfield, K. R. and B. J. Gliddon, Adsorption in Shallow Fluidized Beds ofLarge Particles, AIChE Symp. Sec, 74 (176), 55-64 (1978).

Crank, J.. The Mathematics of Diffusion, 2nd Ed., Oxford University PressLondon, 1975.

Barton, K. C, R. D. LaNauze, J. F. Davidson and D. Harrison, Bubble Growth

Due to Coalescence in Fluidized Beds, Trans, t Chem E 55 274-280(1977). '

—

'

Davidson, J. F. and D. Harrison, Fluidized Particles, Cambridge UniversityPress, 1963.

Davidson, J. F., n. Harrison, R. c. Darton and R. D. Darton and R. D.LaNauze, The Two-Phase Theory of Fluidization and Its Application toChemical Reactors, Chapter 10 of "Chemical Reactor Theory", ed . byLapidus L. and N. R. Amundson

, Prentice-Hall, Englewood Cliffs N J583 (1977)

.

. . .,

Decker, N. and L. R. Glicksman, Heat. Transfer in Large Particle FluidizedBeds, Int. J. Heat Mass Transfer, 26. 1307-1320 (1983).

Donnadieu, G., Heat Transfer in Granular Media - Fixed Beds and FluidizedBeds. Rev. Inst. Franc. PotroleAnn. Combust. Liquids, 16 1330-1356(1961).

~

Uwivedi, P. N. and S. N. Upadhyay, Particle-Fluid Mass Transfer in Fixed andFluidized Beds, Ind. Engr. Chem. Proc. Des. Dev

., 16, 157-165 (1977).

Ergun, S., Fluid Flow Through Packed Columns, Chem. Eng. Prog., 48, 89-94

Fan. L. T., C. c. Chang, Y. S. Yu , T. Takahashi and Z. Tanaka , IncipientFluidization Condition for a Centrifugal Fluidized Bed, AIChE J 31999-1009 (1985)

.

'—

'

11-80

Perron, J. R. and L. L. Watson, Studies of Gas-Solid Heat Transfer inFluldized Beds," Chem. Eng. Prog. Symp. Series, 58 (38), 79-86 (1962).

Frantz, J. F., Fluid-to-Particle Heat Transfer in Fluidized Beds, Chem. Eng.Progr., 57 (7), 35-42 (1961).

Fryer, C. and 0. E. Potter, Experimental Investigation of Models forFluidized-Bed Catalytic Reactors, AIChE J., 22, 38-47 (1976).

Ganzka, V. L., A Mechanistic Theory for Heat Transfer between Fluidized Bedsof Large Particles and Immersed Surfaces, J. Heat Mass Transfer, 25,1531-1540 (1982).

Geldart, D.,Types of Gas Fluidization, Powder Tech., 7, 285-292 (1973).

Grace, J. R.,

An Evaluation of Models for Fluidized-Bed Reactors, AIChESymp. Ser. 67 (116), 159-167 (1971).

Grace, J. R., Fluidized Bed Reactor Modelling: An Overview, ACS Symp. Ser.,168, 3-12 (1981).

Gupta, R. and Majumdar, A. S., Aerodynamics of a Vibrated Fluid Bed, Can. J.Chem. Eng., 58, 332-338 (1980).

Hanni, P. F., D. F. Farkas and G. E. Brown, Design and Operating Parametersfor a Continuous Centrifugal Fluidized Bed Drier, J. Food Scl., 411172-1179 (1976)

.

Heertjes, P. M. and S. W. McKibbins, The Partial Coefficient of HeatTransfer in a Drying Fluidized Bed, Chem. Engr. Sci., 5, 161-167 (1956).

Heywood, B.,

Continuos Vibrating Fluid Bed Drying, Food Processing Ind.,April, pp. 21-25, 1978.

Hoebink, J., Thesis, Technical University Eindhoven, Holland, 1977.

Information Bulletin on Foreign Chemical Industry, U.S.S.R., No. 24/79/52853(1960)

.

Jason, A. C, Effects of Fat Content on Diffusion of Water in Fish Muscle,J. Sci. Food Agric, 16, 281-288 (1965).

Jinesiu, G. I., Equation of the Minimum Fluidization Rate in Condition ofVibration, Rev. Roum. Chim. , 16, 1255-1268 (1971).

Kato, K. and C. Y. Wen, Gas-Particle Heat Transfer in Fixed and FluidizedBeds, Chem. Eng. Progr. Symp. Ser., 66(105), 100-108 (1970).

Kato, K., H. Kubota and C. Y. Wen, Mass Transfer in Fixed Fluidized Beds,Inst. Chem. Engr. Symp. Ser., 66(105), 86-99 (1970).

Kettenrlng, K. N. , E. L. Manderfield and J. M. Smith, Heat and Mass Transferin Fluidized System, Chem. Engr. Progr. 46(3) 139-145 (1950).

11-81

Kobayashi, H., F. Aral and T. Sunawaka, Determination of Gas Cross-FlowCoefficient Between the Buble and Emulsion Phase by Measuring theResidence Time Distribution of Fluid in a Fluidized Bed, Kagaku Kogaku31. 239-246 (1967).

Kroger, D. G. , E. K. Levy and J. C. Chen, Flow Characteristics in Packed andFluidized Rotating Beds, Powder Technol., 24, 9-18 (1979).

Kunii, D. and 0. Levenspiel, Fluidization Engineering, Wiley, N.Y. (1969).

Kwauk, M.,Generalized Fluidization I. Steady-State Motion, Scientia Sinica,

12 (4), 587-612 (1963).

Leva, M., Correlations in Fluidized Systems, Chem. F.ng. , Nov., pp. 266-2701957.

Levy, E., N. Martin and J. Chen, Minimum Fluidization and Startup of aCentrifugal Fluidized Bed, in Fluidization, pp. 71-75,, ed. by D. L.Kearins, Cambridge University Press, Cambridge (1978).

Lewis, E. W. and E. W. Bowerman, Fluidization of Solid Particles in Liquid.Chem. Engr. Prog., 48, 603-610 (1952).

Lewis, W. K., E. R. Gilliland and W. C. Bauer, Characteristics of FluidizedParticles, Ind. Engr. Chem., 41, 1104-1117 (1949).

Luikov, A. V., Heat and Mass Transfer in Capillary-Porous Bodies, PergamonPress, London, 1966.

Luikov, A. V., Theory of Drying, Energiya, Moscow, 1968.

Luikov, A. V., Analytical Heat Diffusion Theory, Academic Press, New York1968.

Lyuboshits, I. L., L. S. Slobodkin, and I. J. Pikus , Drying of DispersedThermolabile Materials, Naukai Tekhnika, Minsk, 1969.

Mathur, A., S. C. Saxena, A. Correlation for Heat Transfer Between ImmersedSurface and Gas-Fluidized Beds of Large Particles, to be presented atthe VIII National Heat and Mass Transfer Conference, Visadhapatman,India, December, 1985.

Mao, Q. M. and 0. E. Potter, Modelling Fluidized Bed Reactors, in 'Frontiersin Chemical Reaction Engineering', Vol. I, ed. by L. K. Doraiswamy andR. A. Mashelkar, Wiley, Eastern Limited, New Delhi, India, 1984.

May, W. G., Fluidized-Bed Reactor Studies, Chem. Eng. Prog., 55 (12) 49-56(1959).

—

McGaw, R. R., The Development of a Mechanism for Gas-Particle Heat Transferin Shallow Fluidized Beds of Large Particles, Chem. Engr Sci 32 Il-ls (1977). '

—

'

11-82

McCormick, P. Y., in R. H. Perry and C. H. Chilton (eds.). "ChemicalEngineers' Handoobk," 5th ed., sec. 20, McGraw-Hill Book Company, NewYork, 1973.

Mori, S. and C. Y. Wen, Estimation of Bubble Diameter in Gaseous FluidizedBeds, AIChE J., 21, 109-115 (1975).

Mrowiec, M. and W. Ciesielczyk, Fluidlzed-bed Dryers for Paste Materials,International Chem. Eng., 17, 273-379 (1977).

Murray, J. D., On the Mathematics of Fluidization. Part 1. FundamentalEquations and Waste Propogation, J. Fluid. Mech., 21, 465-472 (1965).

NASA, Rotating Fluidized Bed Reactor for Space Nuclear Propulsion - AnnualReport, NASA Agreement No. 13254, Washington (September, 1972).

Nelson, P. A. and T. R. Galloway, Particle-to-Fluid Heat and Mass Transferin Dense Systems of Fine Particles, Chem. Eng. Scl . , 30, 1-6 (1975).

Nonhebel, G. and A. A. H. Moss, Drying of Solids In the Chemical Industry,Chapter 2, Butterworth, London, 1971.

Niro Atomizer Inc., Fluid Bed Dryers, Catalog number B-Na-171, Columbia,Maryland (1980).

Orcutt, J. C, J. F. Davidson and R. L. Pigford, Residence TimeDistributions in Fluidized Catalytic Reactors, Chem. Eng. Prog. Sym.Ser. , 58 (38) 1-15 (1962)

.

Osinskii, V. P., B. S. Sazhin and E. A. Chuvpilo, Chem. Pet. Eng. (EnglishTranslation) 11, 866-869 (1969).

Pandey, D. K. and S. N. Upadhyay, Mass Transfer between the Particles andthe Fluid in Fluidized Beds of Large Particles, Int. J. Heat MassTransfer, 24, 1221-1228 (1981).

Partridge, B. A. and P. N. Rowe, Chemical Reacton in a Bubbling Gas-Fluidized Bed, Trans. Inst. Chem. Engi-s . , 44, T335-T348 (1966).

Priestley, R. J., where Fluidized Solids Stand Today, Chem. Eng., 69(14)125-132 (1962).

—

Richardson, J. F. and W. N. Zaki , Sedimentation and Fluidization: Part I,Trans. Inst. Chem. Engrs., 32, 35-53 (1954).

Richardson, J. F. and P. Ayers , Heat Transfer Between Particles and a Gas ina Fluidized Bed, Trans. Inst. Chem. Engrs., 37, 314-322 (1959).

Richardson, J. F., Incipient Fluidization and Particulate Systems, inFluidization, ed. by J. F. Davidson and D. Harrison, Academic Press, NewYork (1971).

Rdman, G. N. , E. Rotestein and M. J. Urbicaln, Kinetics of Water VaporDesorption from Apples, J. Food Scl., 44, 193-197 (1979).

11-83

Rowe, P. N., A Note on the Motion of a Bubble Rising Through a FluidizedBed, Chem. Eng. Sci . , 19, 75-77 (1964).

Rowe, P. N. and B. A. Partridge, An X-Ray Study of Bubbles in FluidizedBeds, Trans. Inst. Chem. Engr., 43, 1157-1175 (1965).

Rowe, P. N., Particle-to-Liquid Mass Transfer in Fluidized Beds, Chem. Eng.Sci. , 30, 7-9 (1975).

Rowe, P. N., Prediction of Bubble Size in a Gas Fluidized Bed, Chem. Eng.Sci., 31, 285-288 (1976).

Saxena, S. C. and J. D. Gabor, Mechanism of Heat Transfer Between a Surfaceand a Gas-Fluidized Bed for Combustor Application, Prog. Energy Combust.Sci. , 7, 73-102 (1981)

.

Saxema, S. C. and V. L. Ganzha, Heat Transfer to Immersed Surfaces in Gas-Fluidized Beds of Large Particles and Powder Characterization, PowderTech. 39, 199-208 (1984).

Schoeber, W. J. A. H., A Short-cut Method for the Calculation of DryingRates in Case of a Concentration Dependence Diffusion Coefficient,Proceedings of the First International Symposium on Drying, SciencePress, Princeton (1978).

Selzer, V. W. and W. J. Thomson, Fluidized-Bed Heat Transfer—The PacketTheory Devisited, AIChE Symp. Ser., 73 (161) 29-37 (1977).

Sen Gupta, A., R. B. Chanbe , and S. N. Upadhyay, Fluid-Particle HeatTransfer in Fixed and Fluidized Beds, Chem. Engr. Sci., 29, 839-843(1974).

Shen, C. Y. and L. T. Jonestone, Gas-solid Contact in Fluidized Beds, AIChEJ. 1, 349-354 (1955).

Shi, Y., Y. S. Yu, and L. T. Fan, Incipient Fluidization Condition for aTapered Fluidized Bed, Ind. Eng. Chem. Fundam., 23, 484-489 (1984).

Shi, Y. and L. T. Fan, Lateral Mixing of Solids in Batch Gas-SolidsFluidized Beds, Ind. Eng. Chem. Process Des. Dev. , 23, 337-341 (1984).

Shi, Y. and L. T. Fan, Lateral Mixing of Solids in Gas-Solids Fluidized Beds'with Continuous Flow of Solids, Powder Technol

. , 41, 23-28 (1985).

Slenov, V. A., and N. V. Mikhailov, Vibrof luidized Bed, (in Russian) Nauka,Moscow (1972).

Steinour, H., Rate of Sedimentation — Nonf locculated Suspensions of UniformSpheres, Ind. Eng. Chem., 36, 618-624 (1944).

Trebal, R. E., Mass Transfer Operations, Third Edition, Chapter 12, McGraw-Hill, New York, 1980.

Twoomey, R. D. and H. F. Johnstone, Gaseous Fluidization of SolidsParticles, Chem. Eng. Prog., 48, 220-226 (1952).

11-84

Upadhyay, S. N. and G. Tripathi, Mass Transfer in Fixed and Fluidized Bed,J. Sclent. Ind. Res., 34, 10-35 (January, 1975).

Vaccarezza, L. M. , Thesis, Facultad de Ciencias Exactasy Naturaies

,

Universidad de Buenos Aires, Argentina, 1975.

Van Deemter, J. J., Mixing and Contacting in Gas-Solid Fluidized Beds, Chem.Eng. Sci., 13, 143-154 (1961).

Vanecek, V., M. Markvart, R. Drbohlav and R. L. Hummel, ExperimentalEvidence on Operation of Continuous Fluidized-Bed Dryers, Chem. Engr

.

Prog. Symp. Ser., 66 (105) 243-252 (1970).

Viswanathan, K., D. Subba Rao and B. C. Raychaudhury , CoherentRepresentation of the Drying of Gas and Solids in Fluidized Beds, IndianChem. Engr., 14, 12-23 (1982).

Viswanathan, K., Semicompartmental Approach to Fluidized Bed ReactorModeling — Application to Catalytic Reactions, Ind. Eng. Chem. Fundam.

,

21, 352-360 (1982).

Viswanathan, K.,

D. Khakar and D. Subba Rao, Fluidized Bed Absorber, Chem.Eng. Commun., 20, 235-251 (1983).

Viswanathan, K. and D. Subba Rao, Drying of Gas and Solids in FluidizedBeds, in Drying '82, ed. by A. S. Mujumda, McGraw-Hill, Washington1984.

Walton, J. S., R. L. Olson and 0. I.evenspiele, Gas-Solid Film Coefficientsof Heat Transfer in Fluiidzed Coal Beds, Ind. Engr. Chem., 44, 1474-1480(1952) .

Wen, C. Y. and Y. H. Yu , A Generalized Method for predicting the MinimumFluidization Velocity, AIChE J., 12, 610-612 (1966).

Yates, J. G., Fundamentals of Fluidized-Bed Chemical Processes,Butterworths, London, 1983.

EMS

APPENDIX A. DERIVATION OF EON. (87)

For the controlled volume element in the bubble phase with a size of

AÂz shown in Fig. 13, a quasi-steady state moisture balance around this

volume element yields

„ f rate of moisture int- by convection

rate of moisture T

out by convection J

rate of moisture in throughgas exchange with the cloud

phase(A-l)

The individual terms in eqn . (A-l) are

rate of moisture in

by convection at z(U A )p xlb b g bl z

rate of moisture outby convection at z+Az

(U A. )p x, I .

b b g bl z-H

rate of moisture in through "

gas exchange with thecloud-wake phase

= (A, A )p (K, ). (x -x. )b z g be b c b

Substituting these individual terms into eqn. (A-l), dividing the resultant

expression by A 4zpp

and letting Az -» 0, we obtain

dxtUv (K. ). (x -X, )be b c b'b dz

This is eqn. (87) in the text.

(A-2)

11-86


Consider the controlled volume element in cloud-wake phase with a size

of A Az represented in Fig. 13, a quasi-steady state moisture balance around

this volume element gives

rate of moisture in

by convecure in "]

tion J"

I

rate of moisture out

by convecure out "]

tion J

rate of moisture in

through gas exchange. with the bubble phase

rate of moisture outthrough gas exchange

. with the emulsion phase .

rate of moisture in

through interphase exchangebetween gas and solids

(B-])

The individual terms in (B-l) are

f rate of moisture in "1

. . . = (u, A e r )o xt by convection atz J be mf g c z

rate of moisture outby convection at z+Az

- (u A e )p x I

J b c mf g el z+Az

rate of moisture in throughgas exchange with the

bubble phase "bVVVVW


emulsion phase(A. A )p (K, ). (x .-x )b z"g be b ej c

11-87

rate of moisture in through '

interphase exchange betweengas and so] ids

(AcV (Wmf>

p Sg P

K (x -x )

Pg P c'

volume volume specific transfer rateof fraction surface of moisture per

cloud of solids of solids unit surfacearea of solids

Substituting these individual terms into eqn . (B-l), dividing the resultant

expression by A.Az and letting Az-fr-0 , we obtainb

A dxc

if dz(Kce). (x ,-x ) i (K ), (x.-x )b ei c ce b b c

(1 - 6_# ) S n K (x -X_)mf P Pg P c

(B-2)

Realizing that

bVb

Vtfh(a+/S)

Vb- (or < fi)

Equation (B-2) now becomes

dx

( ° + « e.f

Ub "di

(K ) (x . - x ) + (K. }(iik-x )ce b ei c be b c

(B-3)

(or + /3)(l-e ) s K (x -x )mf P pg p c(B-4)


II-8

APPENDIX C. DERIVATION OF EQN. (89)

Referring to Fig. 13, under quasi-steady state assumption a moisture

balance around the emulsion gas in ith compartment yields

rate of moisture in "} f rate of moisture out ")

by convection by convection

rate of moisture in

through gas exchangewith the cloud- wake phase

rate of moisture in

through interphase exchangebetween gas and solids

The individual terms in eqn. (C-l) are

rate of moisture in

by convection at theinlet of ith compartment

= {u A e r )p x. ( i

)

e e mf g in

rate of moisture out

by convection at the. outlet of ith compartment

(u A e „)p x .

e e mf g oi


cloud-wake phaseAjz, (z - z. , )p (K ). (x -x .)b i 1-3 "g* ce'b v

c ei ;

rate of moisture in throughi nterphase exchange between

gas and solidsA (z- z. . )p (l-€ „) S K (x -x .)e i-l 'g 1 mf p pg p ei

'

Substituting the individual terms into eqn. (C-l) and dividing the resultant

expression by A (z-z. )p yields

A (x ,- x. , . )

e ei in(i)U 1—— €e A. z - z. , mf

b i-1

A

(1 - f „) S K (x -x .)mr p pg p ei

but

b t b

[l-«b(l«or +0)j

'h

Thus , we have

"-ib'

1^"V,f"er%' i "

[1-* (l+a+/8)]

; (1-e ,) S K (x -x .)6 mf p pg

v

p ei'

11-89

1 ^ceVVei' ( c- 2 »

Ae

Atf]--4

b(l+o+/3)]

(C-3)

+ (Kce>b (VX

ei> ' c-4 >


11-90


The moisture content of particles in the dryer decreases with time due

to evaporation. Applying moisture balance around the entire solid phase in

the bed yields

f rate of accumula- "] r rate ofI tion of moisture J I moisture in

frate of moisture out

I. by evaporation(D-l)

The three terms in eqn. (D-l) are

rate of accumulationof moisture

dx

p dt

frate of "1

_I moisture in J

rate of moisture outby evaporation

rate of moistureadsorbed by the

- surrounding drying gas

= (U AJp (x. - x Jt g in out

Substituting the individual terms into eqn. (D-l) and dividing each term by

M , we obtainP

dx p A—2 -

g ° fc

(x x )

dt M in out'(D-2)


11-91


Recall the bed temperature is assumed to be homogeneous. Applying

energy balance around the entire bed, we obtain

f rateI of

rate of accumulationtherma] energy

rate of thermal energyin accompanied by

inflow drying gas

rate of thermal energyout carried by

exit drying gas(E-l)

Choosing T = °C as the reference temperature, we can write the individual

terms in eqn. (E-l) as below

rate of accumulationof thermal energy C M -rr

p p at

rate of thermal energy I

in accompanied b

inflow drying ga

in accompanied by » (U A, )p [x. (r +c T. ) + c T. 1o t. g in o wg in g in

rate of thermal energyout carried by exit

drying gas= (U AJp [x .(r +c T) 4 c T]

o t g out o wg g'

Substituting the individual terms into eqn. (E-l) and rearrangement of the

resultant expression give

dt

dT Wt ,,"IF [(xin-

Xout>*o

+ Cg(T

in-T)

P P

+ C (T. x. - Tx J]wg in in out

In most practical cases,

|c (T. - T)|»|c (T. x. -Tx )|I g in I I wz m in out I

(E-2)

(E-3)

11-92

Accordingly, eqn. (F.-2) is simplified as

p U A

XT - g„°t

[(x. -x A ) y + c (T. -T)] (E-4)dt M c in ouA "o g in l '

PS 5


11-93


Referring to Fig. 17, the portion of the cloud confined between 6 and

e+A6 has volume

R , 2rr , e+A9

er, o eb

p sine'd6'd\//dp

3 3 re+Ae

it (R - R. ) sinS'de'c d j

e(F-l)

Under quasi-steady state assumption, a moisture balance around the gas cloud

in this controlled volume leads to

=rate of moisture in by

1 f rate of moisture outthrough-flow gas ]-( by through-f low gas

rate of moisture in throughinterphase exchange between

gas and solids

rate of moisture in throughexchange between cloud and

emulsion phase(F-2)

The individual terms in eqn . (F-2) are

rate of moisture in

by through-f] ow gasat e gc g cle

rate of moisture outby through-flow gas

at 6+A9* Wde.A

11-94

rate of moisture in

through exchange betweengas and solids

e mip K (x -x )

e Pg P c'

volume volume specific moisture transferof fraction surface rate per unit

cloud of solids of solids surface area of solids

f rate of moisture in

|through exchange between

I cloud and emulsion phase -

(2;rR )(R de) -p K (x -x ) sine»* \* 3 o c c

interface area transfer rate of

moisture per unitarea in the direction

normal to the interface

Insertion of individual terms into eqn. (F-2) and substituting V from eqn.

(F-l) yield

e+A9= Q

gcUcle " X

c le+Ae» > I " <»c' R

b>< I8ine-de-)(i-e.

f)s

K (x -x ) + (2rrR )K sinede • K (x -x )PEPC cc cec (F-3)

Dividing by A6 sine and letting A6 -*, we obtain

Q 3x

sine ae I n-(R3-Rf)(l-p is -K (x*-x )3 c b at p pg p c

(2irR )R -K (x -x )c c c e c

(F-4)


IMS

APPENDIX G. DERIVATION OF EQN. (154)

Under pseudo-steady state assumption, an energy balance around the

controlled volume depicted in Fig. 17 gives

thermal energy in

accompanied by throughflow gas

thermal energy outaccompanied by through

flow gas

thermal energy in

through exchange betweengas and sol ids

thermal energy inthrough exchange betweencloud and emulsion gas

(G-l)

Choosing T = 0°C as reference temperature, we can express three individual

terms in eqn . (G-l) as

thermal energy in

accompanied by through-flow gas at 8

Q P c T| rtgc g g le

thermal energy out

accompanied by through-flow gas at 0+A0

Q P c Tlgc g g I9+A9

thermal energy throughexchange betweengas and solids

e mlfa (TP s

volume volume specific transfer rate ofof fraction surface thermal energy per unit

cloud of solids of solids surface area of solids

11-96

thermal energy in through '

exchange between cloudand emulsion gas

(2ttR ) (R de) H (T -T ) sinece e c

interface area transfer rate of thermalenergy per unit area in

the direction normal tothe i nt erface

Substituting these individual terms and the expression for VG into eqn. (ti-

ll , dividing each term by sinede and taking A6 - give

QgPgCg dT 2

. as3s - 2ff R B (T -T )sine d9 p ce

v

e c'

| rr (R3

- R3)(t-e |Sh (T -T )3 c b mf p p p c

(G-2)


11-97

APPENDIX H DERIVATION OF EQN . (161)

For the controlled volume with a size of A A depicted in Fig. 18, a

quasi-steady state moisture balance leads to

o . rrate

l in

rate of moisture in byby convection

1 _ frate of moisture out "l

J I by convection J

rate of moisture in through'

exchange with cloud-wake phase

The various contributions to eqn . (H-l) are

b b g bl z

rate of moisture

in by convectionat z

rate of moistureout by convection

at z+Az(u. A, )p x, I

b b g blz+Az

rate of moisture in through'

exchange with cloud-wake phase

(AbAz)

gc'g(x - x, )en b

volume of flow rate of concentration differencebed with through-flow between through-f low gasheight Az gas per unit entering and leaving

volume of bed bu

Substituting these individual terms into eqn. (H-l), dividing each term by

Afa

P Az — and letting Az -*, we obtain

E*b

dXb

b b dz gc en b (H-2)


11-98

APPENDIX I. DERIVATION OF EQN. (165)

Referring to Fig. 16, a quasi-steady state energy balance around the

controlled volume with size of Ab oz is

rate of thermal

energy in byconvection

rate of thermalenergy out by

convection

rate of thermal energy *

in by exchange withcloud-wake phase

The individual terms in eqn. (1-1) are

(A u )p c Tlb b"g g Iz

rate of thermalenergy in by

. convection at z

(I-J)

rate of thermal

energy in byconvection at z+Az .

(A u )p c Tlb b "g g \-ng g Iz+Az

rate of thermal energyin through exchange

with cloud-wake phase .

(AbAz)

nQ Pgc g

(T - T, )en b

volume of flow rate of temperature differencebed associated through-flow between the through-with the c gas per unit flow gas entering and

volume of bed leav bu

Inserting these individual terms into eqn. (J-l), dividing each term by (—

Az co) and letting Az -»- yields

dTb

U.J, —j- " n Q (T - T )b b dz *gcv en V


(1-2)

11-99

APPENDIX J. DERIVATION OF EQN. (187)

Equation (146) can be rewritten as below

3x _

where

gc xEc p

2irR2-K R

3-R

3

a, - fi~" . «. = | ff~- (1-6 _)s k x

,

Q-gC4 3 Q

g0n,f p g p

2ITR2K

a5

= -Q^2 Xe '

y = C0Se (J-2)gc

To solve eqn. J-l, first evaluate integration factor, which is determined by

(or r/1+or ^— - vllJ

-t uj

exp{-J {^[1+^(^1 - y )]

_1+ a }dy}

r 1 , a~ L -m ] 2 • exp (-or y) (j_3)

1 <*/-{ - y)

Multiplying both sides of eqn. (J-l) by the integration factor and rearrang-

ing the resultant expression yields

or.

_i

h {xc

[-

772—

r

1

°2 exp ( -<v>>

1 * a2 (^ y)

a4

! 72 ]

a2'exp (

"aoy)i * *

2<*| - y)

11-100

" as [

" "7?—-1 °2 exp (-<V> ( J ^ 3 >

From eqn. (J- 2), we obtain

Ql

R

o^=

3Sp (J-*)

For spherical particles, <(> = 1 and

o §_ _6p d 4 d ( J_5 )

P s p

Thus

a. R

a2

' 3Sp

" 1 (J-6)

Now eqn. (J-3) becomes

h {xc[lTa

2(f - »» • ^p(-v»»

» - «5U +a

2 (^ - y)] • exp(-a3y) (J _ 7)

Integration of eqn. (J-7) gives

J2xc[lfa

2( 2 y)] '

exp(~a3y)

" (a Q7" [1 a

2(_

2" V)]

2! EXP ! Q

3 y)3 3 or

+ constant

Using boundary condition, eqn. (147), we obtain

/r or a Qpa

cr/-

xbexp( -a

3 2>=

(f+a H?> ex P t-^» >" constant

3 8 of

(J-8)

11-101

/2, ,„ "4 Q

5.Vs.

2a3

constant = expf-a^ ) (x^ — - — + -) (J _9)

Substituting into eqn. (J-8) results in

3 3 a3

Z5-or. a a„orr

+ exp[a3(y -^|)] (x

b- ^ +5§ + -2-|)> (J . 10)

3 3 a3

From eqn. (J-2), we can determine

°4 (Rc-

Rb' * S~ - , - (1 - e -) s x k , — = x

,

°3 3 R2K

*f P P B or

c c

a or (R3-R?) k

2 5 c b g2

* xe 2 ( J-")

cr„ R K R3 C C p

Substituting expressions in (J-13), together with y=cos6, into eqn. ( J-10)

,

and rearrangement of the resultant equation gives

( r3 ~r?> k Kgc p

,Rr

Rb» ( R3 - R f» k AT

c cgc p

< Rc-

Rb»

Spkg . a***- «

3R2

''

<» - e„f

>*p

' «P t- -j^ (f - cose))3R

cKc

gc

(Rr

Rh>

k

c c p

11-102

(R3-Rhs ,

2V (1

"6.f>Vp >}

'(J_12)

3R K B eC C

Substituting S » — into eqn. (J-12) and rearrangement yieldP

/9(e)xc-[/8*+0(e)]x

e+/8*x*

* * *x, - ( 1 +/S ) x + £ xb e p

2n?2

r-

exp [- —— k (— - cose)] (J-13)ge

where

3 3R -R. k —

0(e) = 1 + z* -£—S (i - 6 )gS (if - cose)

gc p

and

3 3

c c p

*Under the assumption x = x , eqn. (J-13) reduces to

x -x 2irR p-

x^r = m) ex» [ ""q^ k

c(i ' cose)1

3 3(R -O k -.

. r, ,„^_>L_I ,4_ cose)fi

Q R ' 2gc p

2ffR2K ,-

C C ,/2e"P[

q(~2 - cose)] (J-15)

gc


11-103

^*coos

1occen

COto

+J 05ai

uc uCD CDx: CO N 0)u CO -H

T-i CH

o AJto - <O) CO

:

CD Cu

2

c;

to

CD cy - c^-o 3 CO 03 Bc CO a> co M t>cd i-i 05 t-t, B .H CO 0)CD o fc» E r-i

<W >> • CJ V0) J-> x: -

02 M CJ us 4-» Jd - aU w w *rt

CU C c +J 51-. H-rt fl -H oc t-, co ro f-. t.,

CL O X cc e:

4)

Q.|

>.

>. P*> >>u bo t- bo (h bfiti ra a bo ca c bo to a toO a «h c C H C -H ae O -P +J -rt O *Ja H CO +> rt gg 4J "i ra

4-> tt to M (- ."0 +J i- S3a CO £1 ra xi v XIcr +-> -H o +J -H CD CO > ti CO > t4 09 > t-.

co

aj

boco4-J

CO

CJ(J-i

CD 4! agj bo boS3.aa CO &0 Eac

(

ci

X3CJ

d) to c•a oq cos

(0 X

11-104

r-t

too>c o o o

CO CO 00 00Oi 0) 05

bJ)

M0) a

(h to 03 !- t. fa fc,

a o) t~ CJ a cuN tO r-i 0) N N N(O *-! c- to t=

cflCO E Oi t> - o £ tfi 6 eai o n oj -i-! O Oi o oi-( +J ri tH o CQ .w +J< - ^ 1) < < <^ - CO

<-t O C -H C 3 o ooj u a co -r-i u 0) {-i u

-H CO O K oH SB X H o CD X SB ~ "z £;

>> >= >> > >> >)^ bo t, to u bo W (h wra c tug co c ra n co c !d e to ca c boc •<-« c e •--. C -M C *1 r-( C q H CO -t-> —i C +J O -P c +j o M W-< o +» -H«H O 4J t- Kj h eg H CO td +J CO -H•P fc, sf +J t-i -^ t. 4J t- +j t-< CO t-i cOra -o +j a ja CO XI CO XI CO XI V CO XI +->

4_> .,_ J-J •,-, *! H O« > b tfi > M > «s > CO > f- •X > ^

(-) ti

C3 CO

t- W T3c c QJ cc ca

S3a.

> 6) tar?

3

COcT

Ac C/3

G co+-J

CJ•h « eu ea -u « S -m Cfl

to >>TJ t-, «

ftl C tf cu a KX O C£ >u t-j

>> i

1 X*J

a JQJ rec T )

CJ : )O a )

:0 X 1

11-105

"—

'

N

a —

Ui i—

i

K 00

i—

i

J3

U 01 h c

m > w > o

11-106

Fig. 1. Single-stage cylindrical f luidized-bed dryer

( Niro Atomizer, 1980 )

Wet Material11-107

Drying qasi

L

v-» •

• • • •

Dry Material

fi•*

V. • T ,'m /•••

\V y»m' • '• A—</

ilar Chamber

Fig. 2. Schematic of a tapered fluidized-bed dryer

11-108

Fig. 3. Two-stage fluidized-bed drying system (Nlro Atomizer. 1 980)

11-109

Fig. 4. Continous multi-stage f luidized-bed dryer ( Toei, 1966)

11-110

.PITOT TUBE

PLENUM

- FLUIDIZEDBED

Fig. 5. Schematic of centrifugal f luldized-bed apparatus

( NASA, 1972 )

11-111

Fig. 6. Vlbro-fluidlzed bed dryer ( Niro Atomizer, 1980 )

11-112

TOPVIEW

SIDEVIEW

Fig. 7. Structural representation of a tapered fluldzed-bed dry

11-113

Fig. 8. Schematic of a section of a centrifugal fluidlzed-

11-114

BUBBLEPHASE

SOLIDS FREE)

<l -«bH I -emf )()-«„>«„,

1-4

Fig. 9. Schematic representation of two-phase theory of

f luidization

11-115

600 -

400

200

1 50 pm copper shot

0.825 MN/m 2|g|

625 pm copper shot

825 MN/m 2(g)

625 (jm copper shot

0.275 MN/m 2(g)

1020 um sand

0.825 MN/m 2(g)

0.5 1.0- 1.5

Fluidizing rate. UIU

2.0

Fig. 10. The variation with gas velocity of the bed-to-surface

heat transfer coefficient without radiation transfer

(Botterill, 1975)

11-116

0.4

0.3

0.2

» 0.1

pe 0.1 0.2 0.3

x = kg moisture/kg dry solid

0.4

Fig. 11. Typical rate-of-drying curve ( Trebal, 198D)

11-117

^ u o, x out.T out

BUBBLE

PHASE

CLOUD-WAKE

PHASE

EMULSIONPHASE

i* 1 th compartment

(SOLIDS FREE) "s

X•1*1

<Kbc ) b %(K ce

, ith>b

:ompartment

} l

1 \

%X

ei

i- 1 th compartment

«b <5 h (a*0) ( 1- V 1 a*/3))

i V

U o. Mn. Tin

Fig. 12. Schematic representation of three-phase and semi-

compartment model for a batch fluidized bed dryer

11-118

Aub

,x b (Z| )

Al

z*4z

BUBBLE PHASE

T(t), x b (z ,t)

Zi-I

ub

,xb

,( 2i)

CLOUD-WAKE PHASE

T(t), xctz.t)

X;Kbc )|

ub - x b

(zi- |

>

ioLiM^VVjlKceX

EMULSION GAST(t), x e ,

(t)

X* t

/ T(t),xp

(t) ^

I SOLIDS /

ub

,xc

( Zi .|) J e ' x in(D

Fig. 13 Moisture transfer between phases with respect to

ith compartment

11-119

1

A x 'n(N)

*eN-I

cin(N- I

)

in(i)

ei- I

in(i- I )

ZN =H,

Z|

Fig. 14. Representation of compartments of upflow emulsion gas

11-120

Fig. 15. Schematic representation of bubble-cl oud model

11-121

1 \

V vI

'EMULSION,

/SOLIDS ,

. T (t) ///

P //

PHASE

GAS

T (tl

x Itl

1

CLOUD

SINGLE

BUBBLE

x (l,t), T.I2.I1b b

Gt IDS

Kcs SOL

Kcp

xc(0,t] T Iftt)

c

; .

'

1f

Fig. 16. Moisture transport in a single bubble, its surround-

ing cloud and in emulsion phase

37T/4

_ Y

Fig. 17. Representation of the exchange zone in cloud

11-122

out' 'out

EMULSION PHASE

x e (t),T e <t)

u e

CLOUD PHASE

xc(0,z,t).T

c(0,z.t)

nQcx b

xb

(Hf).T

b(H

f)

BUBBLE PHASE

Tb(z.t).x

b(z ,t)

ub

z *Az

Fig. 18. MoIsture transfer between phases

CHAPTER III

MODELING AND SIMULATION OF

A CONTINUOUS FLUIDIZED-BED DRYER

III-l

MATHEMATICAL MODELING

A schematic diagram of the model is shown in Fig. 1. The present model

is based on the two-phase theory of fluidization (see, e.g., Davison and

Harrison, 1963). The underlying assumptions of this theory are that the bed is

divided into two phases, a bubble phase and an emulsion phase (which remains

in minimum fluidization conditions), and that the excess flow of the fluidizing

fluid above minimum fluidization passes through the bed as bubbles. The fluid

in the bubble and emulsion phases and the solid particles are considered to

be continua. Additional simplifying assumptions imposed in deriving the

present model are as follows:

1. The bubble phase is solid-free and the size of bubbles is

uniform and fixed at the so-called effective bubble size.

2. The movement of bubbles through the bed is of plug flow.

3. The clouds surrounding the rising bubbles are very thin, and

therefore, the bubble phase exchanges mass and energy only

with the emulsion gas.

4. The emulsion gas and solid particles are perfectly mixed.

5. Solid particles are added and removed at a constant rate.

6. The inlet temperature and moisture content of solids are

assumed to be uniform.

7. The internal resistance of solids to mass and heat transfer

is negligible.

8. Particles are considered to be uniform in size, shape and

physical properties.

9. The temperature and moisture content of each particle depend

on its age, t , that is, the length of its stay in the dryer.

III-2

As a consequence of assumptions 4 and 5, the residence time

distribution function for solids under a steady-state condi-

tion is

1Cs

f(t ) - ^ exp(- 3 (1)

10. Viscous dissipation is negligible.

11. The changes in Che physical properties of both solids and

drying gas due to the change of temperature are negligible.

These assumptions give rise to the mass and energy conservation equa-

tions for each phase of the fluidized-bed dryer.

Mass Conservation Equations

A - Bubble phase . A steady-state moisture balance around the controlled

volume depicted in Fig. 2 gives (see APPENDIX A)

Ubdx

b

b

with the boundary condition:

(2-b)

Integration of eqn. (2-a) , subject to eqn. (2-b), gives|

> f

- (KbeVb,

*b= X

e " (xe

" X 5 exp[ 5 Z] (3)

b

The parameters in this expression are evaluated from the following relation-

ships.

1. The bed fraction of the bubble phase, 6, :

b

\ - 1 -^ (A)

III-3

where H /H is given by (Babu ^t al. 1978)

Hf

14.311(U-U J " 738d

l*°°6p

°- 736

—i- = 1 + ° mf E P ,„H , ,,0.937 0.126 ° ;

mf U _ pmf g

Alternatively,

(Un - U )mrb (U - u ) + u,

mf br(6)

where U is given by (Davison and Harrison, 1963)

Ubr

= 0.711(gdb

)

- 5(7)

2. The superficial gas velocity through the bubble phase, U :

Ub

= U " Umf (8)

3. The minimum f luidization velocity, U _ (Wen and Yu, 1966):mt

d U p / d3p (p -p )g\

' 5

-E^=((33. 7)

2+ 0.0408 ^^L^l\ _ 33 . 7 („

4. The gas interchange coefficient based on volume of bubbles,

(K ) (Kunii and levenspiel, 1969):

<Kbe^ = iTorirTTTor-T- cwjce b be b

where

1/2 1/4u

rD g

^+5.85-^db d

(Kbc'b^- 5 ii+5 - 83 -JS7V- WWb

e ,D u, ...

(Kce)b. 6 , 78(

_-£rf£b)i/2

(12)

Vb

with

III-4

Deff * e

mfDg

(")

e in the above expression can be approximated by (Broadhurst and Becker,

1975),

„ f = °-^ s

-°- 72'

;

""; 3

'°- 029 ^)"

P (P_

P )gd wsg ws *g' fa

p

B. Emulsion gas . From a moisture balance around the entire emulsion gas,

illustrated in Fig. 3, we obtain (see APPENDIX B)

f= (U A )p (x -x ) + / A p (K, ) (x,-x )dzmrtgue i D g ue b d e

+ (H A )(1-S )(l-c )(-f)o(x* - x ) (15)rc b mt d p eP

If we define the average moisture content of gas bubbles, x. . asb

\'tJ \ dz (16)

eqn. (14) can be rewritten as

rrar . .

S Hf6

fc

e

Cl-ef)(l-6. ) ,

eg%sh ("V x

e>+ —

&~ T "<VV < 17 >

b p

The parameters in the above equation can be evaluated from the following

relationships

1. Evaporation coefficient, o, (Palancz, 1983):

III-5

h p D

(18)

whe

h = c Un p j.PrP 8

HgJh g

-2/3(19)

Nu, ,-, „ -0.441- 77 Re if Re > 30

P P -

hRePr 1/3

P 8l

5 .70Re-°-78

if Re < 30P P

(20)

with

Nu =,

P k

h dP P

C„

1J„ d UnP

-i-S, Re = P ° gk ' p (1-e r )p

(21)

2. Average' moisture content of the drying gas on the surface

-*of a particle, x :

x « / -— exp ( -) x" dt(22)

where x may be expressed as (Palancz, 1983)

xp

- *x(T

p) *

2(x

p) (23)

W " °- 622-76CTF (24)

and

III-6

Wn , n

x (x +K)P PC

x (x +K)pc v

p

if x > xP pc

if x < xp - pc

(25)

In eqn. (24)

(0.622 +10

7.5TP

238 + T(26)

and n and k are constants.

C. Single solid particle . A moisture balance around a particle

depicted in Fig. 3, results in (See APPENDIX C)

dx

dt(1 + — x ) — a(x - x )

p pc d P e'w p

(27-a)


po (27-b)

Equation (27-a) is coupled with T since x is a function of both x and tP P P P'

The average moisture content of particles, x , is obtained as

x = / — exp( )-' x dtt

P s(28)

III-7

Energy Conservation Equations

A. Bubble Phase . From Fig. 2, a steady-state energy balance around

the controlled volume gives (see APPENDIX D)

Uh

dih

P JT-T- " (H, ) u (T -T. ) + p (BL )• (x -x. ) i (29)g o, dz be'b e b Kg be b e b we

b

where

i. = c (T.-T J + x, [c (T.-T ,) + Tn ] C 3°)b g b ref T> wv b ret

i = c (T -T ) + Y n (3Dwe wv e ret

From eqn. (3) ,

°Wb<W " Vb'W^'(K

lT"

bz) °2)

Using the above three expressions along with eqn. (2-a) , we can rewrite

eqn. (29) as

dT, T -I. (It ),6, 6, (K, ) (x -xn)c K.b e b be b b b be b e wv , oc, . , ,,, .

-r- = -,

—

r[ —n~ + r. exp(- -rr~6 z) ] (33-a)dz (c +c x. ) U, p U, u b

g wvV b Kg b b

The appropriate boundary condition is

Tu = T at z=0 (33-b)b

(H, K in eqn (33-a) can be determined by (Kunni and Levarspiel,be b

1969)

("be'b= UK ). + 1/(H„) U

(34)b " ^("bc'b

+ 1/(Hce>b

III-8

where

U ,p c (k p c )1/2

1/4

eyb.«.5-û + s.B -^f-» (35)

b db

e u1/2

(H ) - 6 .78(p c k )

1/2 (-Sp) (36)S E

Vb

B. Emulsion gas . Referring to Fig. 3, a steady-state energy balance

around the entire emulsion gas gives (See APPENDIX E)

= (U A )p (i -i ) + (HA)(1-6)(1-e .)-~o(x*-x )imrtgUe rt b mid pe-wc

''„

PgAb°£

be )b

(Vxb)i

„ edz

where

- (H£V(l-6 b

)(l-£mf)fhp

««- V (3?)

p

_iws " «W<VW + ?„ < 38 >

*0= VV1^ + X0l c

„v<T0-T

ref>+ V < 39 >

i = c (T -T ,) + x [c (T -T c ) + Y„] (40)e gv

e ref e wv e ref '0 v'

III-9

1T =

J— exp(- 3 Tdt

t

) 1 _P s (41)

We define the average temperature of gas bubbles, T, , asb

Tb=ir. /Tb

dz

f u(42)

and the specific heat transfer surface of the dryer wall

Sw

aw " V

tot

Insertion of eqns. (38) through (A3) into eqn. (37) yields

p U= ~^~ {c

<>(Tn~T r- f)

+ xnl c(Tn-T r~> + Yj " c (T -T .)H 8 ref wv ref '0 g e ref

x [c (T -T ,) + V„]} + 6, (11 ). (X, -T )e »v l e ref '0 b be b b e

(A3)

+ (1-6, )(l-€ .) -j- (x -x )[c (T -T A + v„]b mf d p e wv N

p refP

a h (T -T ) - (1-6,)(1-e .)— h (T -T )w w w e b mf d p e PP

PB6b(Kbe'b<

xe-

xb^ cwv<TiTref ) + V <W >

-*;=£ {cg(T

e-T ) +(x

e-x )y + c

OT[(VIref )V (VTref

)Kol )

111-10

&h %>Jh<VTJ + d-fiJU-E ,)-f { a(x*-x) [c (T -T .) + Y ]bbebbe b mfdpe wv p ref r n

pep Kg b be b T> e wv e ref '0

+ a h (T -T ) f45 iwww e V+^J

Eliminating the term, (\ e)b (\-\) , from the above expression by resort-

ing to eqn. (17),etjn. (45) can -be rewritten as

-^-^(c +c x.)(T -T )H * g w e

VêVVV + ^^'dfVV^V^+V

a h (T -T )(46)

w w w e

The heat transfer coefficient between air and dryer wall, h , is correlatedw

as (Li and Finlayson, 1977)

¥* o-« ^°-"»7)

C. Single particle . Referring to Fig. 4, an unsteady-state energy

balance around a particle yields (see APPENDIX F)

di p

H W p

III-ll

where

i - c (I -T ,) + x c (T -T ,)p P P ref p w p ref (19)

ws wv p ref (50)

The energy balance around the stagnant film surrounding the particle yields

(see Fig. 4)

q + 0(x -x )is p e we

h (T -T ) + o(x -x )ipep p e ws (51)

q - 0(x -x ) is p e ws

- h (T -T ) - o(x -x )ipep p e we (52)

Insertion of eqns. (19), (52) and (31) into Eq. (48) and rearrangement of

the resultant equation yield

dt dx

p [(c +x c )—^ + c (T -T ,) —£ I

s p p w dt w p ref dt J

(1 + ) -^-{h (T -T ) - a(x -x )[c (T -T J+yJJ (53)ppcdpep pewve ref

dt p ,

p (c +x c )—f-

= (1 + — x ) -f- {h (T -T ) - a(x"-x )*s vp p w' dt p pc' d p e p' p e

s w pv

111-12

s

Eliminating dx fit from eqn. (54) by resorting to eqn. (27-a), weP s

obtain

dt Pft *

p (c.+x c ) -r-2 = (1+ — x ) -.-{ h (T -T ) - 0(x -x )*s p p w dt P pc a l

p e p p es w p

•[c (T -T .) - c (T -T J + v„] }wv e ref w p ref(55)

Y„ in eqn. (54) is to be evaluated at T c . It can be related to the heat'0 ref

of vaporization at any arbitary temperature , T, as follows:

c T ,— c I , + yn \= c T - c T + vj (56)

w. ref wv ref 0. w wv '0'

ref

For convenience, we choose T - 0°C, then we have

cTr-CT

r + Yn l

- Yn I(57)

w ref wv ref '0' '0

T=0<>

cref

Thus, eqn. (55) becomes

dt

p (c +x c ) -rr2

s p p w dt

Pft

- (1+ — x ) -j- [h (T -T ) -a(x -x )(c T -c T +yJ] (58- a)p pc dpep pe v wewp'0


T T „ at t =0 (58-b)p P0 s

and Ynto be evaluated at T=0°C. The average temperature of particles, fuP.

can be evaluated from

111-13

oo 1 Ss!-/«-=- exp(- —) T dtP - - p s (59)

s s

The moisture content and temperature of the outlet gas. * and Tout out'

can be evaluated from moisture and energy balances, respectively:

Unx . = ,x + U. x, (Hr ) (e.r\\out mf e b o £ ^ '

and

U„ [c T + x (c T + Y„)lg out out wv out

= U [c T + x (c T + v„)lmf g e e wv e '0

+ ub

[Vb (Hf) + W<cwW + V ] < 61 >

Rearrangement gives

Xout ^ IT [

Umf

xe+ Vb (Vl («)

and

'-"VVV^ {V^gV+Xe^Te+ V]

.

+ Ub[cgV H

£) +x

b(H

f)(c

wvTb(H

f) +Yo )]

- VoutV (63)

Equations (3), (17), (33-a), (46), (27-a) and (58-a) with the approp-

riate initial and boundary conditions constitute the governing equations of

the present model. To determine the drying characteristics, these equations

need be solved simultaneously. Because of the coupling and non-linearity

amoung them, it is necessary to employ numerical solutions.

111-14

NUMERICAL SIMULATION

The soultion of the model equations is obtained through a two

dimensional trial-and-error procedure. To determine "x and T , whichP P

are characteristic of the requirement of the drying process, we start

with an initail guess on temperature and moisture content of the emulsion

gas, that is, x and T , respectively. The physical constraints, namely,

T „ < T < max(T ,T )pO e w

-0 1 Xe 1W

are helpful in determining the values for the initial guess. Integration

of eqns. (27-a) and (58-a) , subject to eqns. (27-b) and (58-b), gives

x CO and T„(O respectively. Then x , T , and T, are evaluated fromp s p s P P b

eqns. (28), (59), (3), (33-a) and (33-b). With x , T and T known,

Xe

and Te

are ca lculated from eqns. (17) and (46), and the resultant values are

compared with the respective initial guesses. The fact that a set of non-linear

integro-differential equations (integration is involved in detemininp x ,T andP P

T ) is contained in the model renders the procedure cumbersome.b

For simplification, first we seek to reduce these integro-differential

equations to a set of first order differential equations. This is achieved

by introducing three new intermediate 'variables.

-t

Xn

=T~ I

XnexP(^~)

dtP c

s P(64 -a)

* ft

dX x t

jf-= I

2exp(- -S-) (64-b)


111-15

X =P

at t = 0- (64-c)

* ic c

T = — fS

T exp(- —) dt* I P

t .

a(65-a)

dtexp(- —

)

(65-b)


T =P

at t = 0; (65-c)

and

* 1 r

b Hf

b(66-a)

b _ _bdz

" H r(66-b)


T, = at z -b

(66-c)

Now x , T and T. can be expressed asP P b p

111-16

iim X(67)

Jlim T (68)

T = T (691

When t exceeds a certain value, e.g., t , x and T in eqns. (64-a) ands spp MV/(65-a), respectively, remain constant; then, we have

x = — / x exp(- ~) dt1

tP

tS

t t oo r1 fS * / S, 1 r * s

-

J x exp(- ~) dt + —J x exp(- —)dt

t tP

ts s t s

t

x o+ X

P oexp(" ~)

p t =t t =t t1 s s ' s s s

(70)

Similarly,

T = TP P Op

t =t F

s s ' t =ts s

exp (- — ) (71)

Thus, the solution of eqns. (3)., (17), (33-a) , (46) and (58-a) can be

obtained by solving only a set of first order differential equations

along with several algebric equations. The calculation procedure is

described below.

I I 1-17

1. Input data

2. Assume the inital values x' for x and T* for Te e e e

3. Choose t , which depends on the speed of convergence and

usually is in the range of 1/3 t - 2t .

4. Evaluate

TP o

and Tb

c =ts s

z=a r

through eqns. (64-b), (65-b) and (66-b) with corresponding bound-

ary conditions by using the Runge-Kutta method.

_* _5. Calculate x , T and I using eqns. (70), (71) and (69).

6. Evaluate x .and T from eqns. (17) and (46).

7. Compare xg

and Tg

calculated in step 6 with the initially guessed

values x^ and T If they are not identical, determine a new pair

of initial values of x and T and repeat steps 1 through 7.

8. Stop when x' , T' and x ,T are identical.e e e e

The stopping criteria used in the present study are

i .1-4

i . -2x' - x < 10 and T' - T < 10

I

e e|

|e e

]

For illustration, the following data are considered (Palancz, 1983);

U - 1 m s

TQ

- 250°C

x„ = 0.015

P = 1 kg in

P = 2500 kg m~3

p = 1000 kg mw °

M = 2 x 10~ 5kg m_1

s"1

k = 2.93 x 10~ 2J m"

1s"

1°C

_1

111-18

1.06 KJ kg1

°C1

1.26 KJ kg"1

°C_1

1.93 KJ kgX

°C1

w- 4.19 KJ kg

l°C

X

= 3 -12.5 x 10 kJkg

po= 0.35

S= 300 s

= 0.5 tn

c= 0.15m

u= 2 x 10" m

-5 2 -12.10 m s

'pO20°C

0.2

3

lxl0~

111-19

RESULTS AND DISCUSSION

Tables 1 through 5 present the performance characteristics of the dryer

in terms of the average moisture content and temperature of particles at the

exit and the moisture content and temperature of outlet gas. A comparison is

given in Table 6 of the performance characteristics of the dryer under the adiabatic

condition with those under the dryer-wall heating condition. Table 7 compares

the performance characteristics of the dryer based on the present model with

those based on Palancz's model under identical operating conditions.

The average moisture content or temperature of particles at the exit is

related to the inlet-gas temperature in Fig. 5. Figures 6 through 11 show

the effects of various operating parameters on variations of the temperature

and moisture content of a single particle as functions of time. In Figure 12,

the temperature and moisture content of a particle based on Palancz's model are

compared with those based on the present model. The three stages of drying can

be clearly identified in the xÔ and T (O curves in Figs. 6 through 11.

The rather short initial stage of the T (t ) curves, each with a steep positive

slope, involves preheating of a particle, resulting in a sharp rise in its

temperature from the inlet value. This value is lower than the dew-point temp-

erature of the emulsion gas, thereby inducing condensation of moisture on the

particle. This gives rise to a rapid increase in the x (t ) curve, which isp s

immediately followed by a linearly declining section representing the constant-

rate drying period. The corresponding portion of the T (ts

) curve is horizontal

since the temperature of the particle stays at the wet-bulb temperature. The

remaining portion of each of the two curves represents the falling-rate drying

period in which the temperature and moisture content of the particle approach

gradually to their respective values.

111-20

Effects of the Operating Parameters

The performance characteristics of the dryer under various T are given

in Table 1. The higher the temperature of the inlet-gas, the higher the tem-

perature of the gas in the bubble and emulsion phases, thus enhancing

the rate of evaporation. This, in turn, results in an increase in the

average temperature and a decrease in the moisture content of particles

at the exit. Note that the T curve in Fig. 5 with T less than 250°C

has a relatively small gradient with respect to T . This implies that

the dryer is not highly sensitive to the change in T if it is less than

250 C. To prevent burning or cracking of particles, the drying operation

need be conducted within this range, where moderate fluctuations in T

will not cause overdrying.

The influence of the superficial gas velocity on the performance

characteristics of the dryer can be discerned in Table 2. When U

increases, the average temperature of particles at the exit increases

appreciably while their average moisture content reduces sharply.

This can be attributed to the intensified mass and heat transfer among

bubbles, emulsion gas and solids. The expressions of eqns. (18) and (19)

are indicative of a strong dependence of the heat and mass transfer

coefficients between the drying gas and particles on U ; an increase

in UQ

accelerates the evaporation of moisture, thereby quickening the

drying of an individual particle. This can be seen from the fact that the

gradients of the X (tg

) and x (tg) curves in Fig. 6 are substantially

increased in the constant-rate drying period. It is worth noting that

these gradients are not affected as Significantly by . the change in U in

111-21

the falling-rate drying period as they are in the constant-rate drying

period. This phenomenon suggests that the fluidized-bed dryer is effec-

tive in enhancing the drying rate mainly in the constant-rate drying, period.

It is possible that an effective drying, system can be contrived in

which a fluidized-bed dryer precedes a conventional moving-bed or packed-

bed dryer, with the latter drying particles with bound moisture content.

Figure 7 demonstrates the relationship between the superficial

gas velocity and the length of the constant-rate drying period. This

relationship can be roughly approximated by the expression

4 -6.2Unt = 4.8 x 10 es

which should be of practical use in the design of the fluidized-bed

dryer.

The effect of the dryer-wall temperature on the variations of the

moisture content and temperature of an individual particle as functions

of time can be observed in Fig. 8. Naturally, a rise of wall temperature

increases the rate of heat transfer to the emulsion gas. This leads

to an increase in temperature of the emulsion gas, thereby enlarging

the driving force for evaporation of moisture from the particle.

Consequently, the average temperature of particles at the exit increases

while their average moisture content decreases as illustrated in Table 3.

Table 4 gives the performance characteristics of the dryer under

different inlet-gas moisture contents. Increasing x increases only

slightly the average moisture content of particles at the exit. As

can be seen from the x (t ) and T (t ) curves in Fig. 9, the increasep s p s ° '

ln XQ

elevates the wet-bulb temperature (temperature of a particle in

the constant-rate drying period) as we;ll as the moisture content of the

111-22

emulsion gas. The former tends to enhance the rate of drying while the

latter tends to lower it; consequently, the overall effect of x on the drying

rate is small.

The effect of the mean residence time of particles on the dryer perfor-

mance is summarized in Table 5. With bed height fixed, the smaller the mean

residence time, the larger the feed flow rate of solids and shorter the con-

tact time between particles and drying gas. This results in a relatively

low average temperature and high moisture content for the particles at the

exit. As illustrated in Fig. 10, a relatively large feed flow rate of solids

leads to reduction in the temperature of the emusion gas, thereby lowering the

rate of drying of an individual particle.

Table 6 compares the performance characteristics of the dryer under the

adiabatic condition with those under the dryer-wall heating condition. Note

that under the latter condition, the average moisture content of particles

at the exit is reduced while their average temperature is increased. This is

due to the fact that additional heat influx from the dryer wall increases the

temperature of the emulsion gas, thus enhancing the rate of drying of a particle

as shown in Fig. 8.

Comparison with Existing Models

As stated in the introduction, various models have been proposed for the

design of a continuous f luidized-bed dryer. In developing these models,

different assumptions have been imposed for simplicity.

A model suggested by Vanecek et al. (1966) considers solids to be one

phase and drying gas to be the other. Unlike the present model, the effect

of bubbling is not incorporated into it. The design procedure resorts mainly

to the residence time distribution function for solids and the drying curve,

111-23

Xp(t

s) '.°btainable from solving the overall energy balance equation. Some

models (see, e.g., Nonhebel and Moss, 1971) assume that the bed temperature

is uniform and that the exit streams are in equilibrium. These models also

assume that the drying gas remains in one phase; all these models are

essentially empirical in nature.

Several investigators (see, e.g., Kato et al. , 1981) have developed

models which are fairly elaborate in expressing relationships of the mass

and heat transfer between solids and drying gas in different drying periods;

nevertheless, none of these models is sufficiently mechanistic in that the

bubbling behavior is not taken into account.

A mechanistic model proposed by Palancz (1983) gives a comprehensive

description of the heat and mass transfer among gaseous and solid phases in

a continuous f luidized-bed dryer. It is free of the assumptions that the

drying gas is homogeneous and that exit streams are in equilibrium. Palancz 's

model appears to be the only existing model comparable to the present one.

In fact, the present work is an amendment and extension of that by Palancz.

The major differences between the two models are as follows:

1. Palancz's model assumes that specific heat of the drying gas re-

mains constant throughout the entire drying process. In other words,

and

The second expression implies that the moisture content of gas bubbles, at ,

remains constant, which is contradictory to the plug flow postulate for the

bubble phase. Moreover, when moisture evaporates into the drying gas from

solids, an appreciable amount of moisture migrates from the emulsion gas

111-24

to the bubbles; It is not plausible that its accompanying thermal energy

can be neglected. A consequence of this assumption is that in Palancz ' s

model, the energy conservation equation for the bubble phase, which corresponds

to eqn. (33-a) , is linear and only contains the first term on the right hand

side of the equation. Subsequently, in his energy conservation equation for

the emulsion gas, the term designating the energy transfer accompanied by the

evaporation of moisture contains only T instead of f -T . This means that the

energy conservation equation depends on the choice of reference temperature,

which is impossible.

2. To evaluate the equilibrium moisture content of the drying gas on the

surface of a particle, Palancz 's model resorts to the approximate expression

*p

* - vy vywith

and

i(y = °- 622

76o^-

vy

xn(x

n+ K)

P P

if x > xp c

if x < xp - c

Note that a discontinuity occurs at x =x in the expression for ()>„(x );P P c 2 p

this is illogical. In contrast, the corresponding expression of the present

model, eqn. (23), does not contain such a discontinuity.

The yy and yy curves of the present model are compared with

those of Palancz's model in Fig. 12. The values of T (t ) and x (t ) of thep s p s

111-25

latter obviously are much higher than those of the former. As mentioned

earlier, the latter neglects the net outflow of moisture from the emulsion

phase to the bubble phase and its accompanying thermal energy transfer.

This is tantamount to including extra mass and thermal energy in the emulsion

gas in establishing mass and energy balances around it. As a result, relatively

high values of x and T are expected as shown in Table 7, which in turn leads

to an overestimation of the values of x and T .

P P

NOTATION

A cross-sectional area of the bed, n

2

111-26

aw

Cg

cp

cross-sectional area of the bubble phase, m

specific heat transfer surface of the dryer wall, m

specific heat of dry gas, kJ kg "C

specific heat of particles (dry basis) , kJ kg "C

c specific heat of water (liquid state), kJ kg °C

c specific heat of water vapor, kJ kg "Cwv

D diameter of bed column, mc

2 -1D molecular diffusion coefficient of the drying gas, m s

2 -1D ,, effective diffusion coefficient of Che drying gas, m sgeff

d effective bubble diameter, mb

d particle diameter, mP

-2g gravitational acceleration, n s

Hf

expanded bed height, m

H , bed height at minimum fluidizing conditions, mmi -

(H ), volumetric heat transfer coefficient between the bubble andCe b

-1 -3 -1cloud-wake regions based on the volume of bubbles, J s m °C

(H, ), volumetric heat transfer coefficient between the bubble andV b-1 -3 . -1

emulsion phases based on the volume of bubbles, j s m C

(H ), volumetric heat transfer coefficient between the cloud-wake region"-1 -3 . -1

and the emulsion phase based on the Volume of bubbles, J s m C

heat transfer coefficient between the drying gas and solids,

J s-1 m"2 "Cr 1

heat transfer coefficient between the drying gas and the dryer wall,

J a-I m- 2 "Crl

enthalpy of Inlet gas (dry basis), kJ kg

enthalpy of gas bubbles (dry basis), kJ kg

enthalpy of the emulsion (-as (dry basis), kJ kg

enthalpy of water vapor on the surface of a particle, kJ kg

hP

111-27

Xws

average enthalpy of water vapor on the surface of particles, kj kg"1

^weenthalpy of water vapor contained in the emulsion gas) W-kg

1 enthalpy of a particle (wet basis), kJ kg

j Colburn- factor

^bc-'bcoefficient of gas interchange between the bubble and cloud-wakeregions based on the volume of bubbles, s

_1

^be^ coefflclent of gas interchange between the bubble and emulsionphases based on the volume of bubbles, s

^Kce^bcoef£ici-ent of gas interchange between the cloudrwake region and theemulsion phase based on the volume of bubbles, s

k^ thermal conductivity of the drying gas, J m"1

"C_1

Le

Lewis number, dimensionless

Nm

Nusselt number, dimensionless

P Prantle number, dimensionless

Pw

pressure of saturated water vapor, mm Hg

qs

conductive heat flux inside a particle, J s_1

m~2

Re

particle Reynolds number, dimensionlessP

Sw heat trans£er surface area of the dryer wall,

TQ

temperature of the inlet gas, °C

T^ temperature of gas bubbles, °C

Tfa

bed-height average temperature of gas bubbles, °C

T^ temperature of the emulsion gas, °C

T temperature of a particle, °C

TP

average temperature of particles, °C

Tp0

temperature of inlet particles, °C

Tref

reference-state temperature, °C

Tu

dryer-wall temperature, °C

t time, ss

tg

mean residence time of particles in the dryer, s

111-28

UQ

superficial gas velocity (measured on an empty bed basis) througha bed of solids, m s~l

Ub

superficial gas velocity in the bubble-phase, based on totalcross-sectional area of the bed, m s

-1

Ubr

linear velocity of a single bubble, m s"1

Umf

superficial gas velocity at minimum fluidizing conditions, m s_1

V volume of the bed, m

xQ

moisture content of inlet gas (dry basis), dimensionless

^ moisture content of gas bubbles (dry basis), dimensionless

^ bed-height average moisture content of gas bubbles (dry basis),d imen s ion less

xe

moisture content of the emulsion gas (dry basis), dimensionless

xp

moisture content of a particle (dry basis), dimensionless

xp

average moisture content of particles (dry basis), dimensionlessA

xp

moisture content of the drying gas on the surface of a particle (drybasis) , dimensionless

average moisture content of the drying gas on the surface of a particle(dry basis), dimensionless

Xp0

micsture content of inlet particles (dry basis) , dimensionless

xpc

critical moisture content of a particle (dry basis), dimensionless

z elevation, m

GREEK LETTERS

Yq heat of vaporization, kJ kg

&h

fraction of the fluidized bed consisting of bubbles, dimensionless

Ee

void fraction in the emulsion phase, dimensionless

Cmf

vold £racc ion at minimum fluidizing conditions, dimensionless

Mg

viscosity of gas, kg m s_1

P density of gas, kg m

Pw density of water, kg m

Pws

density of wet solids, kg m

a evaporation coefficient, kg m

*s

sphericity of a particle, dimensionless

111-29

LITERATURE CITED

Babu, S. P., B. Shah and A. Talwalkar, Fluidization Correlation forCoal Gasification Materials - Minimum Fluidization Velocity andFluidized Bed Expansion Ratio, AIChE Symp. Ser. 74, 17 6-186(1978).

Broadhurst, T. E. and H. A. Becker, Onset of Fluidization and Slug-ging in Beds of Uniform Particles, AIChE J., 21, 23 8-247(197 5).

Davidson, J. F. and D. Harrison, Fluidized Particles, Chapter 1, pp.19-20, Cambridge University Press, 1963.

Kato, K.,

S. Omura, D. Taneda, I. Onozania and A. lijima, DryingCharacteristics in a Packed Fluidized Bed Dryer, J. Chem. Eng.Japan, 14, 36 5-371(1981).

Kunii, D. and 0. Levenspiel, Fluidization Engineering, Chapter 7,Wiley, New York, 1 969.

Li, C. H. and B. A. Finlayson, Heat Transfer in Packed Beds - A Re-evaluation, Chem. Eng. Sci. 38, 147-153(1977).

Nonhebel, G. and A. A. H. Moss, Drying Solids in the Chemical Indu-stry, Chapter 11, Butterworth, London, 1977.

Palancz, B., A Mathematical Model for Continuous Fluidized-bed Dry-ing, Chem. Eng. Sci. 38, 1045-10 5 9(1983).

Vanecek, V., M. Markvart and R. Drbohlav, Fluidized Bed Drying, Tran-slated by J. Landau, Leonard Hill, London, 1966.

Wen, C. Y. and Y. H. Yu, A Generalized Method for Predicting theMinimum Fluidization Velocity, AIChE J. 12, 610-612(1966).

111-30

APPENDIX A. DERIVATION OF EQN. (2-a)

Consider Che controlled volume element with a size of A Az shown in

Fig. 1. A steady-state moisture balance around this volume element gives

= (rate °f moisutre in) _ (rate of moisture out]V by convection / V by convection /

/rate of moisture in through \+ (gas exhange with the emulsion]

(A-l)^ phase

The various contributions to the moisture balance are

rate of moisture in] .

by convection at z/ ^b'v pg

Xb ' z

rate of moisture out\

by convection )= (U A ) n x[

at z+Az /b cg b'z+Az

rate of moisture in\through gas exchange] =

(A Az)p «) ( x -x )with the emulsion / ° g be b e b

phase '

By substituting these expressions into eqn. (A-l) , dividing the resultant

expression by Afa

Azp and taking the limit as Az goes to zero, we obtain

^ô*" *'"* '-'^'W (A- 2)

The expression within the braces on the left-hand side of eqn. (A-2) is the

first derivative of x with respect to z; thus,

Ub

dxb

67^ = (Kbe )

b(VX

b) <A" 3 >

b

This is eqn. (2-a) in the text.

111-31


Under che assumption of complete mixing, the moisture content of the

emulsion gas is constant throughout the emulsion phase. At steady state,

a moisture balance around the emulsion gas gives

rate of moisture in

by convectionrate of moisture out

by convection

rate of moisture i

through gas exchangewith the bubble

phase

rate of moisture in

by evaporationof moisture in solids

(B-l)

The individual terms in eqn. (B-l) are

rate of moisture inby convectionat z=0

(U CA ) p x„mf t

Jg

rate of moisture out'

by convectionat z = H r

(U J\. )p xmf t

/hg e

rate of moisture in

through gas exchangewith the bubble

phase

rate of moisture inby evaporation

of moisture in solids

VgêVVe'^

01fV

totalvolume

of the

bed

(1-5.) (l-£ ,)b mf

P

a(x -x )

P e

volume ' specific averagefraction surface evapova-of solids of solids tion rate

per unit

surfacearea ofso lids

Substituting these Individual terms Into Eq . (B-l) gives

"VfVVW^ JO

fAbPg(Kbe )b<Vx

e)d »

+ (HfAt). d-8b

)(l-emf

) .(^)o(x„-xJhp


111-32

(B-2)

111-33

APPENDIX C. DERIVATION OF EQN. (27-a)

The moisture content of a particle in the dryer decreases with time

due to evaporation. Applying moisture balance around the particle gives

/rate of accumulation

\ of moisturenl = /rate o f moisture rate of moisture \

out by evaporation/ (C-l)

The three terms in eqn. (C-l) are

rate of accumulation)of moisture / ^

pc

dx

dt

volume density ratio of rate ofof of wet mass of changea particles a dry of moisture

particle particle content ofto that a particleof a wetparticle

rate of moisture

race of moisture ouCby evaporation

(TTd )0(X -X )

P P e

The density of wet solids, p , is related to Che moisCure conCained

in the void of solids, M , and Che mass of dry parcicles, M , through:

M + M

M H~=/ s, M

= (—

)

sVs V

wV.,sMw1+—

-

M= p*

ss VMM

1w s w

J.

M V Mw s s

(C-2)

-

111-34

M

Ms

p M- s w

p M

(C-3)

The moisture content in excess of x exists only as a liquid filmpc J M

surrounding solids and thus should not be considered as a part of the

water contained in solids. Therefore , we have

M

and correspondingly, eqn. (C-3) becomes

1+xE5

ws s p

i+ ?~xpc (C-5)

Substitution of the three individual terms into eqn. (C-l) and rearrange-

ment of the resultant equation give

P, ,. - - (1H x )—— a(x -x ) (T-Ms dt p pc' d p e \i* v)

This is eqn. (2 7-a) in the text.

111-35


For Che controlled volume element with a size of AÂz shown in Fig.

2, a steady-state energy balance leads to

('rate of thermal energy

in by convectiony) /rate of thermal energy]

/ y out by convection /

/rate of thermal energy \ /

i in through bubble-emulsion] + /

\ interphase exchange / \

rate of thermal energyin accompanied by net moisture influx at bubble-emulsion interface

(D-l)

The individual terms are

rate of thermal energyin by convection at z

'rate of thermal energy

out by convection at

V z + Az

'rate of thermal energy

>

in through bubhle-emulsion interphase

^ exchange *

'rate of thermal energyXin accompanied by nfet \

moisture influx at J

bubble-emulsion interVface

(UbV Pg^lz

(UbVVbUz

(V2)p8

(Vb (xe - \> L

ve

By substituting the above expressions into eqn. (D-l), dividing the

resultant expression by A, Azp and taking the limit as Az goes to zero,

we have

ub

dib

* -^~ = (Hu K (T - T, ) + p (K, ), (x - xj i0. dz be b e b g lie b e T> w

(D-2)


JII-36

APPENDIX E. DERIVATION OF EON. (37)

Referring to Fig. 3, a steady-state energy balance around the entire

emulsion gas is

rate of thermalin by convec

energy)

tion /rate of thermal energy]

out by convection /

rate of thermal energyin by evaporation ofmoisture in solids

fate of thermalenergy in fromthe bed wall

rate of thermal energytransfer to solids

rate of thermal energyin through bubble-emulsion interphase ex-

change

rate of thermal energyout accompanied by mois

ture flux at bubbleemulsion interface

(E-l)

The various contributions to eqn. (E-l) are

rate of thermal energy]

in by convection at z = 0/(U CA )p . i„v mf t'*g

rate of thermal energyout by convection at

z = H

(UmfA)p - irat t g e

rate of thermal energyin by evaporation of

moisture in solids

(H,A ) - [(1-6 )(l-e J] *

(-f-)it b mr aP

0(x - x )

total volumevolume of fractionthe bed of solids

specificsurface of

so lids

averageevaporation-

rate per unitsurface area

of solids

averageenthalpy

of moisture

on thesurface ofso lids

111-37

change

rate of thermal energyin from the bed wall

rate of thermal energyout accompanied by mass

flux at bubble-emulsion interface

/rate of thermal energyI transfer to solids

/ VHbeVW dz

S h (T -T )w w w e

Ja5p (K, ), (x -x, )i dzn d g be b e b we

(HfAt

)

totalvolume

of bed

[(1-6 )(1-e )] •

(-f)• h (T -T )

b rat d pepvolume specific heat transfer

fraction surface rate per unitof solids of solids surface of

so lids

Substituting these individual terms into eqn. (E-l) yields

o = (u Op (in-i ) + (H,Aj(i-6.)(l-ef ) 4-aCx* - x ) iraftgue ft b mfd p ewe

P

£+

J Vb.WV** +WW- J V^beVV^)i dz

we

(H A )(l-6 )(l-£ ) ~- h (T -T )rc b mtdpepP

(E-2)

This Is eqn. (37) in the text.

111-38


An energy balance around a particle in Fig. 3 yields

(rate of accumulationôf thermal energy J

= /rate of thermal energy] /rate of thermal enI in by conduction / I out by evaporati

energyon

(F-l)

The individual terms are

(

rate of accumulationof thermal energy )

ird 1+x

(PPC

s P„

p pc

<l£->

di_Edt

volume ofa solidparticle

density ofof wet

particles

ratio ofmass of a

dry particleto that of awet particle

rate ofchange

ofenthalpy

of a particle

rate of thermal energyin by conduction K>q

»

te of thermal energy\ „

3ut by evaporation / = (ird )o(x -x )i/ p p e ,P p e ws

Substituting these individual terms into eqn. (F-l) and rearrangement of

the resultant equation give

di p

P -r-2 = -f-(l+ —x )[q - a(x - x )i 1

s dt d p pc lHs p e us

s p w(F-2)


111-39

Table 1. Performance characteristics of the dryer under various T(

(T =105°C, T n-20°C, Un=lm/sec. , x n-0.35, X.-0.015, T =300sec)w pO pO s

T [°C] Te[°C] %'">

*P["

J fp[°C] T

brd x [-]

out' 't r°c]out

50°C 45.9 0.055 0.250 44.8°C 46.7 0.054 45.9

100°C 51.6 0.068 0.218 50.2 61.3 0.067 51.8

150°C 58.0 0.085 0.195 56.4 76.5 0.084 58.4

200°C 64.8 0.092 0.166 62.9 91.9 0.091 65.5

250°C 72.0 0.100 0.143 69.9 107.7 0.099 72.9

Table 2. Performance characteristics of the dryer under various U

(T =250°C, T =20°C, T =50°C, x ,,=0.35, x -0.015, T =300sec. )pu w pO s

U [m/sec.

]

TeCC] x

e ["J -pi") fpm y-a W-l T (°C]

out

0.80 51.0 0.075 0.271 49.9 85.5 0.060 51.8

1.0 55.3 0.075 0.199 53.8 94.4 0.074 56.2

1.2 63.7 0.075 0.153 61.9 105.5 0.074 64.6

111-40

Table 3. Performance characteristics of the dryer under various Tw

(T =250°C, Tft=20°C, U =lm/sec. , x ^0.35, x =0.015, T =300sec.)

pO pO s

yc] Te[°C] y-] V" 1 t

p['C] y°ci x [-1Out 1 '

T f°C]out

50 55.3 0.075 0.199 53.8 94.4 0.074 57.1

75 63.0 0.092 0.174 61.2 100.5 0.090 63.9

105 72.0 0.100 0.143 69.9 107.7 0.099 72.9

Table 4. Performance characteristics of the dryer under various x

(T =150°C, T =250°C, T =20°C, U =lm/sec, x n=0.35, T =300sec.

)

po pO

x [0] Te[°C] y-i V" 1 T

prc] T,rc] x [-]

out ' t r°c]out

0.015 72.0 0.100 0.143 69.9 107.7 0.099 72.9

0.050 75.5 0.135 0.148 73.4 112.5 0.134 76.6

0.100 79.6 0.180 0.150 77.5 118.6 0.179 78.1

IIT-41

Table 5. Performance characteristics of the dryer under various ts

(T-250-C, Tw=105°C, T

p0=20"C, Lylm/sec. , X

pQ=0.35, x =0.015)

ts[sec] T

ercj x

e[") y-> T

p[»C] y-ci x [-]

out '

To»t l ,C ]

150 57.9 0.093 0.239 55.5 96.5 0.092 58.8

300 72.0 0.100 0.113 69.9 107.7 0.099 72.9

4 50 86.0 0.090 0.092 84.3 118.9 0.089 86.8

Table 6. Comparison of performance characteristics of the dryer under adiabaticcondition with those under bed-wall heating condition.

(T =250°C, Tp0=20°C, U

Q=1 m/sec. , x

pQ=0.35, x =0.015, T

s=300sec.)

Case rwrc] yci «.H v- ] y-c] y°c) x [-]

out 1 ' WC '

Adiabatic 58.0 58.0 0.083 0.193 56.4 96.5 0.082 58.9

Bed-wail Heating 105.0 72.0 0.100 0.143 69.9 107.7 0.099 72.9

IUr-42

Table 7. Comparison of the performance characteristics of the dryer based on presentmodel with those based on Palancz's model

(T =250"C, T^SCC, T =20°C, » =Wset. , x ,,=0.35, x =0.015, ts=300sec.)

Model Te["C] "J"! V" 1 T [-CI

PTb[-] !WtH T [°C]

.Out

Present 55.3 0.075 0.199 53.8 9*.« 0.074 56.2

Palancz's 66.3 0.159 0.228 66.3 102.8 0.158 67.2

111-43

SOLIDS INPUT

OUTLET GAS

! f

i A

h

(Hbe>b

w

EMULSION(Vb

BUBBLEPHASE PHASE

>

<—SOLIDS OUTPUT

INLET GAS

Fig. I. Schematic diagram of continuous drying

in the fluidized bed.

111-44

o' out ' out

U -,T ,xmf* e' e WS"1

.1

EMULSIONPHASE

COMPLETEMIXING

(Hh »kbe b

(Kbe'b

BUBBLEPHASE

T(z),x L (z)

£ Z + Az

U„,T„O' o

Fig. 2. Mass and energy transfer between the bubbles

and emulsion gas.

111-45

*-> Te

mf e e

i

\SOLID

\ PARTICLES /

\ /

EMULSION GAS

(KlJb

IH ).

be b

BUBBLEPHASE

mf oo

Fig. 3. Mass and energy transfer between solid particles

and emulsion gas.

111-46

o~(x*-x )i

p e we

EMULSIONGAS

h (T -T )

P e p

*—

;

STAGNANT FILM

SOL/

Fig. 4 Energy balance around the stagnant film surrounding

a solid particle.

111-47

0.30

0.20

0.10

0.0 1

70

lt-a

60

50

Curve Variable

V's'

V's'

5040

100 150 200 250

Inlet-gas temperature, T [°c]

Fig. 5. Effect of the inlet-gas temperature : T =105 'C

Tpo

= 20-C, U = lm/sec, xp<)-0.35 . x =0.015,

Ts=300 sec.

300

80

70

60

50

40

30

20

10

\.

r\

\ 1.2

^T\\

.^ 1.0

\N

N>_0.8

\

I^

\

01s

1.0

1.2

Curve Variable

Tp

(ts )

xp

(ts )

Parameter: U [m/sec.J

J_

100

time, t

200

[sec]

300

111-48

0.4

0.3

— 0.2

0.1

0.0

400

Fig. 6 Ettect ot the superficial gas velocity: TQ

250° C

,

Tn =20°C, T =50°C , x n =0.35, x =0.015, i = 300 sec.V0 w ^0 ° b

111-49

0.5 1.0 1.5

Superf ical gas velocity , U I s e c .

I

Fig.7. Plot of length of the constant-rate drying period against

the superficial gas velocity:

xpo=0'35

-xo

T =250°C, T =20°C,T =50°C,o po w

0.015, ts=300 sec.

90

30

20

10

Curve Variable

TP «s>

*p('s>

Parameter: Tw "C

111-50

0.4

0.3

0.2 -2

0.1

50 100

time, t [secj

150 200

Fig. 8. Effect of the dryer- wall temperature: T =250°C,

T =20°C.U =lm/sec, x = 0.35, xQ=O.OI5, t =300 sec.

90

111-51

0.4

30

20

10

0.100

0.3

0.2

Curve Variable

Tp <»s>

*p

(ts

)

Parameter: X [—

J

j L J L_L20 40 60 80

Time, t s [secj

100

Fig. 9. Effect of the inlet-gas moisture content.- T =I05°C,

T =250°C , T =20°C, U =lm/sec, x =0.35,o p o Po

7S

= 300 sec.

20

10

Curve Variable

Tp »s 1

X (t )

Parameter : 7 [sec]

50 100

time, t [sec]

150Jo200

Fig. 10. Effect of the mean residence time of particles:

T =250°C.Tw =l05'>C.Tpo = 20 ,>C, U =lm/sec,

*Po

= 0.35 , x =O.OI5.

90

-80

30

20

10

Curve Case

1 With bed- wall heating

2 Adiabatic

Curve Variable

P s

V's>

J

0.4111-53

0.3

0-2

0.1 *

50 100

time.t [secJ

150 200

Fig. II. Comparison between adiabatic and bed-wall

heating cases: T =250°C , T"w

= I05°C , T = 20°C

UQ

= I m/sec, x =0.35, xo= 0.OI5,"^ = 300 sec.

c

90

40

30

20

10

Curve Model

1 present

2 Palancz's

Curve Variable

vvyv

111-54

0.4

03

0.2 •=

— 0.1

I 150 100

time, ts [sec

J

150 200

Fig. 12. Comparison of the present model with Palancz's model:

T =250 C,Tw=50 C,Tpo=20 C,U =l m/sec, x

p=0.35,

xQ=0.015, t

s= 300 sec.

CHAPTER IV

CONCLUSIONS AND RECOMMENDATIONS

IV-1

CONCLUSIONS AND RECOMMENDATIONS

An extensive and critical review of the f lu idized-bed drying has

been performed. It has given rise to a fairly rigorous mechanistic

model for a continuous f luidized-bed dryer. Based on the model, the

numerical simulation has been conducted for the purpose of investiga-

ting the influence of the various operating parameters. The results

indicate that the performance characteristics of the dryer are affec-

ted significantly by the superficial gas velocity, inlet temperature

of the drying gas, mean residence time of the solids and dryer-wall

temperature .

In drying, the moisture content of the drying gas is appreciably

increased by evaporation of moisture from the solids. Consequently,

a substancial amount of energy is transfered through this moisture

migration. This factor is taken into account in the present model.

In contrast, the existing comparable model proposed by Palancz

assumes that the specific heat of the drying gas remains invariant.

Results of simulation have proved that this assumption leads to an

over-estimation of the temperature and moisture content of the

solids.

It is unlikely that the moisture content of the drying gas on

the surface of a particle can undergo a discontinuity as suggested

by the existing model. The present model does not contain such a

discontinuity, and thus is more rational in expressing the heat and

mass transfer relationships between the drying gas and solids.

In the modeling, it has been assumed that cloud does not exist

around the bubble. This is often true in the case where bubbles are

small as they leave the distributor and move slowly to the top. When

IV-2

the bubble has grown sufficiently large to have cloud, three-phase

model should be employed to include the transport processes occuring

in the cloud. It is worth pointing out that even though particles in

both the emulsion and cloud-wake phases are commonly assumed to have

residence time distribution of the completely mixed type, they should

be treated differently because of their different moisture contents

and mean residence times.

When small bubbles leave the distributor, they tend to grow ra-

pidly by coalescence as they rise through the bed. Consequently, hu-

midification of the bubble gas may occur primarily in the lower

portions of the bed. It would be of interest to couple the bubble

growth model (with changing bubble size) with the two-phase model

employed' in the present work.

Results of numerical simulation show that the f luidized-bed

dryer is effective mainly in the constant-rate drying period. Thus,

it appears advisable that a f luidized-bed dryer be used in series

with a conventional moving-bed or packed-bed dryer; the latter serves

to dry particles with bound moisture content.

-

MODELING AND SIMULATION OF A CONTINUOUS FLUIDIZED-BED DRYER

by

YIMING CHEN

B.E. ChE. Zhejiang University, Hangzhou, China. 1982

AN ABSTRACT OF A MASTER'S THESIS

submitted in partial fulfillment of the

requirements for the degree

MASTER OF SCIENCE

Department of Chemical Engineering

KANSAS STATE UNIVERSITY

Manhattan, Kansas

1986

An extensive and critical review of the f luidized-bed drying has

been performed. It has given rise to a fairly rigorous mechanistic

model for a continuous f luidiz ed-bed dryer. The model depicts the

interactions between gaseous and solid phases in detail. Performance

of the dryer has been simulated numerically based on the model. The

effects of the operating parameters on performance characteristics

of the dryer have been investigated. These include the superficial

gas velocity, inlet temperature and moisture content of the drying

ing gas, mean residence time of the solids and dryer-wall tempera-

ture. Results of numerical simulation indicate that the performance

characteristics of the dryer are affected significantly by the super-

ficial gas velocity, inlet temperature of the drying gas, mean resi-

dence time of the solids and dryer-wall temperature. These results

have also been compared with those based on an existing model. The

comparison shows that the present model is a significant improvement

over the the existing model.

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